Should I fill this gap between floor boards and wall?
It will do no harm to fill the gap. You may find that there are vermin proof/resistant foams on the market. If you use an aerosol dispensed foam, you can place the tip of the dispensing straw to stick a little onto a vertical surface below the floor and let that cure. Then using the cured foam as a shelf to hold the next application of foam to fill the remaining gap.
It will help reduce drafts too.