Are we allowed to send Pokéstop Submissions to Niantic?

This form has now been taken down. I can see no substitute at this time, and I believe Niantic is not currently accepting new Pokestop submissions.

The below answer - in its original form - is obsolete.

You can submit new Pokestops using this form. Word of caution, while it is an official form, this is almost definitely subject to change, and is probably not going to be a stable link for long. It also may take quite a long time to process, given how much load they're probably under. (Someone poke me if this link no longer works.)

All Pokestops and gyms come from Niantic's previous game, Ingress, where users submitted tons and tons of places to become portals in that game. Niantic repurposed the Ingress Intel Portal Map (you must have an Ingress account to view) for use in Pokemon Go.