"... and all our yesterdays have lighted fools the way to dusty death." -- Shakespeare's Macbeth.
In my experience, "yesteryear" is a very, very rarely used word. I remember as a small boy that there was a series of toy cars called "Models of Yesteryear". That's the only place I remember ever seeing the word used. I have never seen or heard "yesteryears", in the plural, that I can recall.
"Yesterday" and "yesterdays" are sometimes used as poetic ways to say "the past". But this is rare. If you are writing a poem or a speech that is supposed to sound profound, it can be appropriate. But I wouldn't use it in casual conversation. That is, "Oh, how we all long for that forgotten yesterday when the hills were green and the sky was blue." Fits. "This corporation did many government projects yesterday." No. People would understand you to mean literally, the day before today. And if you made it clear, "In a bygone yesterday this corporation did many government projects" ... the tone is not appropriate for a business conversation. It MIGHT work if you are speaking whimsically, but it would only work to the extent that people understood that you were deliberately using an odd tone for effect.
"Yesterdays", plural, is doubly poetic, and I would avoid that even more outside the context of poetry or rhetoric.