Sealing gaps to prevent pest ingress
Probably not.  My flat has a balcony with a couple of wooden cupboards built in to one side.  Wasps have built a nest in one of those (despite there not being much of a gap).  Obviously, if you can make the shed air-tight, that will stop them, but otherwise - not so much.

To deal with wasps:

Firstly: can you live with them?  They are an important predator of garden pests, so ideally just leave them alone.  However, if you use the shed, this is probably not practical.

If you find the nest when it is small (upto 10 cm), blast it with a few seconds of a wasp-killer late one evening, and the nest should be dead a day later.  

If you find the nest when it is large (upto 30 cm), same process except a) empty the full can on the nest; b) wait until a cool evening if possible; c) dress up with clothing covering tightly as much of your skin as possible (this will be hot), and prepare to run directly away if the wasps start to leave the nest.  I did this when the evening temp was about 20C without problem.

(In other words, you almost certainly don't need a pest removal firm in future).

Finally:


Don't do what my significant other did, and try to dig a wasps nest out of the ground at midday.
If your significant other does this and gets stung, "Well what did you expect?" is not the correct response.