Not if woody portions of the grapevines weave from one side of the fence to another through the chain link voids.

Thanks Larry. I'm not sure, but I'm guessing this is the case. We may be able to leave that one side of fence till a later date. In this case, What would the proper timing & procedure for trimming it back?

Thanks for the advise!!

I grow several grapevines up against chain link (hurricane) fencing. It
is a great trellis. BUT you need to keep the vine's trunk on one side
of the fence and I enforce this with winter annual pruning. If you let
the vine grow through and stay it will expand eventually stretching
(damaging) the fence and girdling (damaging) the vine from that point
upward.

Unless someone did this you can almost assuredly expect the vine grown though the fence possibly at several places.

Things you can do:
(optional) If there is part of the vine running along the ground
(particularly if the growth is not yet woody) you can cut and remove a
half section of bark and bury it in a pot. It will root. Kind of like
doing an air layering. Eventually you cut the vine from the mother
plant. Then you have a potted grape vine that can be transplanted to
the new desired area.

(optional) You can take dormant cuttings, callous them and then plant
them either directly in the ground or pots, again to produce a new vine.

Either preserves the vine in case something goes wrong.

As to the old vine and the fence: The vine is likely fairly woody so it
will stand on its own once the fence is removed. But it is not used to
being unsupported so it will be somewhat weak. If you do not mind
cutting up the fence in that area where it is attached (it can be weaved
back together with minimal loss of length later if desired) and
assuming the worst kind of entanglement between vine and fence then it
should be a breeze to do what you want. I'd wait until the leaves are
gone so you can better see the issues. And being South Dakota, I'd wait
until winter is pretty much over but the vine has not broken dormancy
to remove the fence. Wire cutters for the fence; I like the wire
cutting notch in pliers for this job. Cut parts of fence can be twirled
up unwinding the cut wire.

May want to do SOME pruning first to get rid of some of the vine that is
in the way. And to cut as many of the tendrils wrapped around the
fencing s you can reach. BUT do keep in mind there will be parts of the
vine you want to keep and the manipulations you do may break some of
the desirable parts necessitating secondary backup parts. So leave more
vine on than you might otherwise in case some gets damaged in removing
the fence. Pruning and metal cutting will be back and forth until the
fencing is finally clear. If you have an old pair of pruning shears you
may want to use them as cutting tendrils results in a few times where
you are actually "cutting" the fence with is really bad for the shears'
edge.

Once the fence is out of the way evaluate as to how much support is
needed. Also look for girdled parts of the vine. Either do more
pruning or bring the new trellis into position. Attach trellis to
ground. Use nursery tape to tie the vine to the trellis. And start
pruning for that years production and next year's wood. Once the vine
breaks dormancy the tendrils will secure the vine and eventually the
nursery tape can be removed.

https://www.sdstate.edu/ps/research/viticulture/grapeproduction.cfm

Hack it off to a stump(s). remove fence, replace with a kniflen two-wire trellis, You'll loose a year of fruit, but grape needs to be cut back every ten to twenty five years anyways.