You can see the original food stall that sold them in a city museum. There, you will notice that the menu simply says "croquette", not jinja croquette. Unfortunately, the food stall closed in 2006.
Here's the good news. The Jinja Croquettes made by Murasakian came in second in the 2016 Isesaki Gourmet Grand Prix. So, here you are. You can also go to one of the city's events like Hatsuichi Market in January, Isesaki Meisen Day (Meisen is the fabric Isesaki is famous for) in March, or JA Chokubaijo (JA farmer's market) to look for them. Check the website for the City's tourism association.
You might want to try making them yourself if you live far away from Isesaki. The ingredients are flour, potatoes, and dried laver seaweed. Some use okara (tofu refuse), too. The firm texture that characterizes the Jinja Croquettes comes from kneading the potatoes well. Also, letting the raw croquettes sit for a while before deep-frying them helps create the thick texture.
You use sauce when you eat jinja croquettes. Children like to use sweet miso bean paste. By using sweet miso, you can create a taste similar to Yakimanju roasted buns.