"I would like to apply on 6 & 7 june, would "it" be available these dates?"?

If you use "it" then it needs to be clear what you are referring to. It is not clear in your sentence. I think what you are trying to say is:
I would like to make an application for June 6th or 7th; would those dates be available?

I need an application form. I would like to apply on June 6 or 7. Would it (would one) be available on those dates?

That could work. In absence of context, though (the two sentences that you have written), there is no indication of what "it" is supposed to be. could be a pen, a desk, a place to write, your eyeglasses, who knows? If you want people to understand what you mean, you have to tell them at one point or another.

would i be able to apply these dates