Q.Star Jasmine Vine not growing
We planted white star jasmine in tall pots to trail up our pergola (permanent outdoor installation in Houston, TX). The first year it did well, but this spring/summer the growth has been very minimal and the vines have fewer leaves. The pots are about 3 feet tall and 1 foot square. The root ball was installed in the top 12″ of the pot for max height onto the pergola. I put an upside down plastic container in the bottom along with drilling a hole for drainage. The rest is filled with soil. I water regularly but allow the soil to dry out between. Do I need to fertilize? If so, with what and how often? Is a permanent installation in a pot not a good idea for this plant?

Yes, your vine does need regular fertilizer application.
These links will help you.
https://www.gardeningknowhow.com/ornamental/flowers/jasmine/how-to-fertilize-jasmine.htm
https://www.gardeningknowhow.com/ornamental/flowers/jasmine/star-jasmine-as-ground-cover.htm