I am looking at a trip to Alaska early this fall and would like some recommendations for long distance lenses that will not break the bank.

Thanks for any advice.

Have you used "long" lenses before?  How big is your bank?  How strong are your arms?

I picked up a Sony 100-400 GM for £1400 2nd hand from MPB.

I used to have a longer zoom from Sigma - much cheaper, but I didn't like the build quality or AF.

Unlike the Sony 100-400mm, Vivitar, Cosina, and Promaster made a 100-400mm (they are actually the same lens) that you can get for under $100 used.  Tokina made a great 135-400mm that's a little more expensive and harder to find, but has a tripod socket.

I was in Alaska back in 2018 and brought my A57 and exclusively used my sony 18-250.  Pictures turned out amazing!  We have about a dozen of them framed and hung around our house.  

One key thing to watch our for is the weight of the lens. I use a Tamron 150-600 G02 (via an LA-EA5 adapter) on my A7iii and that zoom range makes it very versatile and It does get me great results. The downside is that at 2.5 kg it's a bit of a beast to use hand-held for any length of time and so I would want to have a sturdy tripod with me - which is yet more weight to carry though. If you do go for a lens with a maximum focal length of 400mm then consider a x1.4 teleconverter, it adds very little weight and makes a real difference.

I have a few of the new generation Tamron lenses. If I went on a similar trip and wanted to go light weight, I would bring my Tamron 17-28 f2.8 and the Tamron 70-300 F5-6.3. They are both fantastic lenses and full frame. If I wanted something longer, I would switch to APSc mode in my camera and the 300mm would become 450mm. So in essence I would be covered from 17mm all the way up to 450MM. That should be enough for most situations. If I needed something wider or longer, then I am shooting for a specific assignment or specialized application. In that case you have to bring more lenses which defeats the purpose of being lightweight. 

You can get Sony lenses if you want. However, they are at least double the cost and 50% or more heavier for marginally better performance. I did a lot of research and actually tried the Tamron lenses before I commited to Tamron as my only lens lineup. I am totally happy with the quality, cost and weigh savings the Tamrons offer me. Syou can also look into Sigma lenses. They are just as heavy as the Sony lenses but cost less.