While against the policies listed for Feed Providers on RadioReference is it technically legal to broadcast an FRS or GMRS feed online?
Nothing illegal about streaming online FRS, GMRS, MURS, CB, Amateur Radio, public safety / land mobile, marine, rail road, shortware broadcasts.
If you can pick it up on a scanner or shortware radio, you can stream it online legally, except for cell phones and encrypted communications.
(It is illegal to merely
possess a scanner that can receive cell phone transmissions.)
So, yes, your idea would be perfectly legal.
Disclaimer: IANAL and this is not legal advice