Jtdylktuy

Are commoners i.e, people not wearing yajnopavita able to attain moksha without needing to be born as Brahmin according to Hindu Scriptures? Are there any examples of them attaining Moksha or Oneness with God(Brahman) or Oneness with their Ishta Devata without being born as Brahmin?

Edit:
Can they(commoners) attain Moksha in their current life?

Yes, anyone can attain moksha regardless of caste, creed and gender. Krishna Himself explicitly states that Shudras, Vaishyas and even women can attain Him:

For finding refuge in Me, O Arjuna, even those who maybe of sinful birth, women, Vaishyas and similarly Shudras, even they can reach the Supreme Goal. (Bhagavad Gita 9.32)

...people not wearing yajnopavita...

This is wrong premise. Dharma and Moksha are independent although dharma secures Moksha. As an analogy any message signal when transmitted over a longer distance it is modulated with career signal because message signal is prone to distortion along the way. So, all these samskArs, varNashram dharma etc are nothing but career signals which are in no way decides moksha, but are traditions which passes the message to the next generation and rest of it. So, the part of casteism is related to dharma not Moksha. Moksha transgresses any border of conditioning, birth, nationality etc.

Are there any examples of them attaining Moksha or Oneness with God(Brahman) or Oneness with their Ishta Devata without being born as Brahmin?

The words GOD or Moksha or Oneness are my pet peeves, nevertheless would say Nisargadatta Maharaja was self realized person who was non-brahmin. He was an Advaitin belonged to nAth Sampradaye. He is very influencial at least according to me. This is what he said

My Guru ordered me to attend to the sense 'I am' and to give attention to nothing else. I just obeyed. I did not follow any particular course of breathing, or meditation, or study of scriptures. Whatever happened, I would turn away my attention from it and remain with the sense 'I am'. It may look too simple, even crude. My only reason for doing it was that my Guru told me so. Yet it worked!

Heis one among many; Lao Tsu, Mansur al haq, Ekhart Tolle etc are other examples. Btw, I found this one quote of him very fascinating.

I am not a woman, I am not a man

neither a saint not a god

I am not the body, I am not the mind

I am not a person of any kind

Neither the self, nor the soul

but much much deeper than the black hole

I am nothing but I am everything

Because everything is nothing and nothing is everything.

But personally I won't recommend you to be Advaitin without any guru from perfect unbroken lineage otherwise you will end up superimposing all the glories of self upon the mind. But if you have Advaitic intuition in the sense you are capable to indwell in the self or say in 'being' than 'becoming' you can start practice although defective it may be.

My Guru told me: "...Go back to that state of pure being, where the ‘I am’ is still in its purity before it got contaminated with ‘I am this’ or ‘I am that.’ Your burden is of false self-identifications—abandon them all." My guru told me, "Trust me, I tell you: you are Divine. Take it as the absolute truth. Your joy is divine, your suffering is divine too. All comes from God. Remember it always. You are God, your will alone is done." I did believe him and soon realized how wonderfully true and accurate were his words. I did not condition my mind by thinking, "I am God, I am wonderful, I am beyond." I simply followed his instruction, which was to focus the mind on pure being, "I am," and stay in it. I used to sit for hours together, with nothing but the "I am" in my mind and soon the peace and joy and deep all-embracing love became my normal state. In it all disappeared—myself, my guru, the life I lived, the world around me. Only peace remained, and unfathomable silence. (I Am That, Dialogue 51, April 16, 1971)

Wait... How could I forget legendary Gautama Buddha!?

There are different views regarding this and also difference concepts of Moksha (Merging with the Self, meant for the jnanis) or Mukti (Salokya, Sajujya,Sarupya, Samipya etc for the Bhaktas). And there are concepts like Sadyomukti (Instant Liberation), Kramamukti(Gradual Liberation), Jivanmukti(Liberation while alive) and Videhamukti (Liberation after Death).

According to some sints like Sri Sitaramdas Omkarnth,non-brahmins can not attain Moksha. They need a brahmin birth for that.They refer to the incident of Srimabhagavatam, Sri Krishna gave darshan to Muchukunda in Narayana form, but told that he would have to do severe ausrerities to remove sin and would get Mukti in the next birth being born as a brahmin.(Reference:Sitaram Leela Lekhya, Promod Ranjan Gupta, page155-6).

But devotees can get Mukti by grace of God any time. In adhyatmaramayana(Lankakanda,11/87),Narada says whateven sinneres get videhamukti,ie mukti after death, if he or she can die remembering God constantly.(Ref :lbid page 156).

Sabari, Jatayu, kavandha etc got sajujyamukti and sarupyamukti by grace of Sri Ramachandra, as referred to in Adhyatmaramayana Aranyakanda, chapter 8,9 and 10
(lbid page 156).

According to Sri Ramakrishna also, pure devotion can beget Mukti for avSudra as well as a brahmin.(Ref:Sri Ramakrishnakathamrita, Udbodhan Akhanda Edition,page 620 and 743).

Yes, Moksha(Salokya, Sameepya, Sarupya etc., and Kaivalya) can be attained by anyone who desires it, regardless of caste or gender

The_Complete_Works_of_Swami_Vivekananda/Volume_7/Conversations_And_Dialogues/

Thus the talk went on, and gradually drifted to the topic of
Shankaracharya. The disciple was a great adherent of Shankara, almost
to the point of fanaticism. He used to look upon Shankara's Advaita
philosophy as the crest of all philosophies and could not bear any
criticism of him. Swamiji was aware of this, and, as was his wont,
wanted to break this one - sidedness of the disciple.

Swamiji: Shankara's intellect was sharp like the razor. He was a good
arguer and a scholar, no doubt of that, but he had no great
liberality; his heart too seems to have been like that. Besides, he
used to take great pride in his Brahmanism -- much like a southern
Brahmin of the priest class, you may say. How he has defended in his
commentary on the Vedanta - sutras that the non - brahmin castes will
not attain to a supreme knowledge of Brahman! And what specious
arguments! Referring to Vidura1 he has said that he became a knower
of Brahman by reason of his Brahmin body in the previous incarnation.
Well, if nowadays any Shudra attains to a knowledge of Brahman, shall
we have to side with your Shankara and maintain that because he had
been a Brahmin in his previous birth, therefore he has attained to
this knowledge? Goodness! What is the use of dragging in Brahminism
with so much ado? The Vedas have entitled any one belonging to the
three upper castes to study the Vedas and the realisation of Brahman,
haven't they? So Shankara had no need whatsoever of displaying this
curious bit of pedantry on this subject, contrary to the Vedas. And
such was his heart that he burnt to death lots of Buddhist monks -- by
defeating them in argument! And the Buddhists, too, were foolish
enough to burn themselves to death, simply because they were worsted
in argument! What can you call such an action on Shankara's part
except fanaticism? But look at Buddha's heart! Ever ready to give his
own life to save the life of even a kid -- what to speak of
"[(Sanskrit)]-- for the welfare of the many, for the happiness of the
many"! See, what a large - heartedness -- what a compassion!

Disciple: Can't we call that attitude of the Buddha, too, another kind
of fanaticism, sir? He went to the length of sacrificing his own body
for the sake of a beast!

Swamiji: But consider how much good to the world and its beings came
out of that 'fanaticism' of his -- how many monasteries and schools
and colleges, how many public hospitals and veterinary refuges were
established, how developed architecture became -- think of that. What
was there in this country before Buddha's advent? Only a number of
religious principles recorded on bundles of palm leaves -- and those
too known only to a few. It was Lord Buddha who brought them down to
the practical field and showed how to apply them in the everyday life
of the people. In a sense, he was the living embodiment of true
Vedanta

One who knows the Brahman is the true Brahmin or Brahmrishi