I used to operate a small buiseness doing just this... capture and relocation of raccoons and skunks.
The easiest way for you to get rid of your tennants on your own and rapidly (and cheaply) is to go buy these items :
Liquid amonia (sold as floor cleaner)... take the nasty smelling stuff ...NOT spring freshness.
Rags that hold lots of liquid
rope
1 broom handle.
then do this :
tie a rope to a corner of a rag
imbibe it with amonia
using broom handle, push the rag as far the burrow, entrance as possible, make sure rope sticks out (like a tampon)
do this to every access point around the shed
Every 2 days (if warm dry weather...everyday), pull on the ropes, re-soak raggs a,d stuff them back under the shed.
This fould the air in the burrow (kinda smokin' them out) and animal(s) will flee/relocate themselves. One way to see if animals are still around is to lightly stuff entrances with dead leaves...if entrance is disturbed, you still havve tennants.
ok, but what to do when shedd is critter free ?
Get wire mesh (1" welded squares...NOT chicken wire). 2 feet wide by as long as you'll need to go around the sides of the shed. This is a bit labour intensive BUT YOU'LL NEVER EVER GET CRITTERS UNDER THERE AGAIN.
Dig a trench 1 fooot down from the shed wall and 1 foot out...all around. Using a forens, wood slat etc... secure the mesh... it must go down from the wall and, at bottom of trench, skirt outward. Done ? ok, fill everythingh back in with CLEAN earth and sand...no big rocks.
The reason is that a critter coming back will try (try) to dig under the mesh to get back in... it'll go down, down, hit the skirt and waste nighttime...never finding the hedge. If you put big rocks or bricks, you'll simply guide the critter right to the hedge.
not protecting the shed will mean new tennants for sure.
Capture and relocation is a rapid fix but you displace the problem. and as I said, the presence of the critters are a symptom...the shed's accessibility is the problem
If you go for livetrapping...best raccoon bait is sardines in oil. Carefull... a trap baited with this will also catch a skunk... and you need to be polite with those

.
Hope this helps
pm me if you need more info.
Yann