Unfortunately the GTEx images are generated on demand then erased later. In case it helps, these are dynamically generated by R from the track data. We can help with finding the exact parameters if you are interested.
The temporary nature of these images is noted by the 'trash' directory they exist in that's part of their URL:
For example that link no longer works. Another option if you have only a finite number of genes you are interested in you can save the images locally.
If you have any follow up questions, our public help desk can always be reached at genome@soe.ucsc.edu. You may also send questions to genome-www@soe.ucsc.edu if they contain sensitive data. For any Genome Browser questions on Biostars, the UCSC tag is the best way to ensure visibility by the team.
I guess the suffix of the URL has some sort of session id instead of gene name so basically to automate it you need to capture that session id.
I am sure there should be some API to process the data throughput programming interface.