Hi Elizabeth!  That's awesome that you are going to Basque country...it's such a beautiful place and the food is so wonderful!  
Unfortunately, you will mostly be drinking Txacoli by the glass in tapas bars so it's hard to know who the producer is.  But honestly I wouldn't worry about it.  Even the real stuff is mostly pretty ordinary. Think of it as a light, great, refreshing accompaniment for the tapas.   You can also try the beer, cider or some red wine, but Txacoli has always been my favorite! In a typical Basque tapas route, you are only having one or two small "chiquitos" of Txacoli with your pintxos in each bar before moving on to the next, so you get to try a lot of different ones!
There are a few good producers but its rare to see their wines in the tapas bars.  Txomin Etxaniz is a larger producer and you may see there wines by the bottle around.  It's pretty good. My favorite producer is Itsasmendi.
I hope you are going to San Sebastian as well as Bilbao and plan on getting out there for lots of tapas.  There are so many of Spain's top restaurants in Basque country, but the tapas or pintxo culture is so much more a part of everyday Basque life.
Keep an eye out on the blog...I plan to do an article on my favorite tapas places in St Sebastian, including a couple of places that have very decent wine!
I hope you have a great time and I'd love to hear about your Txacoli adventures!