Carol, there are plenty of shared hosts that can run Moodle 3.4, but issue remains with moodle adding to its requirements (that once use to be recommendations in earlier versions) therefore making shared hosts not meeting every requirement as name suggests, in shared hosting you are sharing the hardware as well as software configuration, so just making changes that your application requires may break it for everyone else.
There are PLENTY of hosting providers that give option of multiple versions of PHP, but when comes to database requirements to convert tables to Barracuda from default Antelope format, this is where all shared hosts will give up.
Though still 3.4 can be run, if you (rather starting from 3.4 directly as new install), start from 2.7, upgrade to 3.0 and then to 3.4 then it wont take database conversion as strict requirement but as recommendation only - make sure your language is set to UTF8 (and everyone even on shared hosting has this capability to change their database languages)