If I primarily bank online, is it a bad idea to open an account with a bank that has no local branch, just in-network ATMs?

Yes. For certain things, such as getting a certified check, you need to see a human employee and cannot use an ATM or do it online. (You could open an account at a local bank for those things, and another account at another bank for other things.)

If your primary bank is online and requires you to have a local branch account, it's not a good choice of online bank. With online banking, make sure you have recourse for cashing checks via ATM or cell phone (pictures). Being able to deposit cash isn't that important but may also be a requirement for you. Most banks that specialize in online services like USAA offer ATM free reimbursement within reason (like 8 transactions a month or something like that).

If you do open a local branch account (this is a good idea) make sure it charges zero fees for any kind of service and a reasonable minimum balance. You also have to remember to access the funds periodically lest the bank claim ownership of the assets in an "abandoned account".