How do I get vinyl tile off of plywood flooring?
Vinegar is not going to be effective at remove the tiles, but what is effective is a bigger tool.  Since you are looking to save money, you might be able to get away with using a cheap scraper.  It will look like a putty knife but a lot stiffer and have a wider blade.  Then using a hammer (wear gloves and glasses), just start tapping the scrapper under the tiles to get larger pieces up.

Once these tiles are up, you will need to get your sub floor (the cheap plywood) to be smooth.  If you got nail pops, rough areas from glue or gashes these will need to be leveled before you put the vinyl down.  Vinyl is not very forgiving in hiding defects.