Can a contact form stay beside the page while it scrolls?

Yes. On a page in the grid layout, place the block (a form, a button, a short note) in the right-hand or left-hand columns, beside the first block it should sit next to. Open its Config tab and turn on Stay In View While Scrolling.

The block becomes a side column from its own row to the end of the page, and it stays on screen while a visitor scrolls. It stops just under your header. If your header hides while scrolling, the block moves up into that space and drops back down when the header returns. The blocks under it make room for it, so nothing has to be resized by hand. On a phone there is one column, so the block goes to the end of the page. Keep it short: a side column taller than the window scrolls inside itself.