Ricky wrote:

...they will
toddle right back...



Notwithstanding the legality issue, it is possible to take them far
enough away that they won't toddle back.

However, the woodchuck population is like a gas (albeit in two
dimensions). If you remove (by any means) a woodchuck, the population
will expand to fill the available space. If you add a woodchuck, the
population will contract, but in either case you have a uniform density
of woodchucks. Of course this does not happen instantaneously, but at
some speed characteristic of woodchucks (might be a few days) having to
do with territoriality. The only way to create a hole (or gaseous
rarefaction) is to increase the heat locally for woodchucks by
installing a dog, bombing their burrows regularly, or otherwise abusing
them on a regular basis. The hole will exist as long as you put enough
energy into maintaining the high temperature.

The same comments apply to squirrels, mice, voles, japanese beetles, and
other mobile pests.