How do I use lemmas that are probably already proven, but including the proof myself is quicker than looking up a source? Suppose I have proven several lemmas during my work that are neither original nor significant (these results were needed for applied engineering/CS research, not mathematics). I believe that if I spend enough time going through articles in minor journals/theses in theoretical CS/applied mathematics, eventually, I will find the statements/proofs of these results. However, it seems that finding these results may take more time than it took to prove the results.
I am curious whether there is a chance that not providing citations could be perceived as plagiarism in one form or another. Would it suffice to state that I do not claim that the results are original without providing a citation?
You do need to make some effort to find in the literature a result you think is known, but after that what you propose is common and I feel is fine.  I do it sometimes.  In such a situation, I tend to say ``the following is probably known, but we include a proof for completeness.''
Even if you find the all your background lemmas in the literature, it might still make the reader's life easier if you include some or all of their proofs.  It is really annoying to have to request seven obscure papers by inter-library loan, and then figure out all the different authors' notation, just to fill in a few pages of proofs.