How impactful can a city be to a startup?

Well I think you just answered your own question sortof
The website is location based, meaning location being one of the most important factors
That said, this is highly subjective. There are benefits to big cities and small.

Big city benefits of course is access to users, and vendors; when you start a site like groupon, even if you're a sales superstar, it'll be difficult getting businesses on board. If you don't have enough businesses represented, you're not really going to get many users, because they will not yet realize much benefit. Bigger cities solve this, due to access to many more opportunities to represent businesses, thus increasing your chance at signing. The disadvantage is the fact that you're not the only one that realizes this, so competition will be a factor, it'll become a money game, who can sign more sales reps, and get to a business quicker.

A smaller city has a benefit of lower competition, however again there are lower resources. For a bootstrapped startup, I'd suggest smaller cities, hire local reps on "generous commission" basis. Additionally, offer businesses finder's fees.

Good Luck!!