I think it might also be acceptable to say 'De frågar efter bröd' ..but I'm not really sure.

Definitely think Ber om is better, but my guess is frågar efter isnt wrong either. A quick Google turned this up: Våga fråga efter en doggy-bag. Which means dare to ask for a doggy bag. My Swedish is not great but I grew up in a Finlands Svenska household and can swear Ive heard this construction used by my parents many times. Very open to being corrected. I should honestly just ask them.

Why not "för" for "om" ? Ber+om sounds too much like "beromd" in a sentance which someone like me would miss hear and think they're honouring bred?

Can anyone explain when to use ber (om) and when to use frågar?

I dont think so . Since fråga is also a noun (question). Frågar is just used when asking a question