Attempted to connect to a Tor bridge (obfs4) using bridges.torproject.org but none of the bridges would connect. Would the next step be to request private bridges by e-mail? Also why does the tool only accept e-mail requests from Riseup or gmail? It's difficult to obtain an invite to get a Riseup account and gmail is not a good way to receive the bridges.
Would the next step be to request private bridges by e-mail?
Well you have two options to get bridges:
email@example.com Gmail or Riseup.
Use MOAT to fetch bridges from the Tor browser. If you want to know how to use MOAT, check out: https://tb-manual.torproject.org/bridges/.
Also why does the tool only accept e-mail requests from Riseup or gmail?
The Tor Project answers this here: https://tor.void.gr/docs/bridges.html.en :
You'll need to send this request from a Gmail, Riseup!, or Yahoo! account, though — we only accept these providers because otherwise we make it too easy for an attacker to make a lot of email addresses and learn about all the bridges.
Also, I do not believe Yahoo is used anymore, given that the main Tor project website does not include Yahoo as an option anymore.