I would say no. You would at least need shell access and the ISP's TOS should allow you to run other programs which most web hosts don't.
Better get a VPS if you want to run a full node.
HTH
If you use a VPS you for sure could. Cheapest is if you do linux on most. Windows on some has extra fee's for license.
But yes you can get a pretty cheap VPS and make a full node. You also could run one at home on a raspberry pi or something. Overtime a raspberry pi likely is to be cheaper as so little electricity vs vps costs.