What zone? How long are shoots right now? Post a photo.

Yes you can prune some now and thin out excess clusters. Then next winter clean things up right.

I would absolutely break off the excess numbers of new shoots. Depending on your local fungus situation you can probably trim your 1+ year old wood but the cut ends will drip sap and could potentially be a source for fungal entry/infection. That is one of the reasons for late winter pruning. Absolutely thin the fruit. In general you want 15-22 leaves per cluster so frequently you will thin to have 1 cluster per shoot (frequently you get 2 clusters per shoot). You can also cut off the bottom 1/3 of the cluster to increase fruit size. Total number of clusters per plant depends on your canopy. Rough rule about 5 per foot of canopy up to about a maximum of 80 for very vigorous vines. These are just ballpark rules I have read about so you need to experiment on your own.

If you are not in a crazy hot/humid area (fungal heaven) I would probably cut back the vines aggressively leaving 50-60 shoots with 1 cluster per shoot. If you are in hot/humid I would probably just decide which 50-60 clusters you want to keep and then remove the shoots that shade those fruit. Then next winter you do your big hack and slash job to remove around 90% of the vine. Many good pruning videos online. Start with something like this:

Grape Video 32 Pruning Neglected Grapevines

https://www.youtube.com/watch?v=PKmN59GZhaQ

If you have wasp/bird depredation consider bagging your clusters to protect them. I used organza bags with excellent results.

Thank you for the responses!! I am in Zone 7, high desert climate. The actual soil around the vines doesn't get much sun, it is against a wall with a large-ish arbor 5ft tall, 5ft deep, 6ft wide. Surrounded by a not-very-healthy lawn. It has been HOT here in Albuquerque so the vines are growing/developing even more rapidly now.

The birds eat plenty but there was more than enough for everyone last summer. (It's my second summer in this house.) My first winter here, the previous owner pruned the vines for me but I couldn't make heads or tails out of how he did it.

I will post photos tomorrow, thanks again!!!