They would just serve the meal to you. You'd be seated at your own table and would get everything that the other tables get. I think a dinner show would actually be more pleasant to do solo than a regular restaurant, as you'll be entertained during the meal instead of dining in silence.
That said, most people at Hoop Dee Doo are there as a family or couple, so if you're self-conscious about being different (at a table alone), that could be problematic.
But there will be no logistical issues, and no one except you would care that you're there solo.
You might get some good natured teasing. During out 2nd trip I saw a single woman sitting and eating and one of the performers sat down next to her and asked her if she needed a husband for the trip out west. She told him yes and he ran as fast as he could from her.
You will be served the same way as everyone else- just not filled to brim and you get your own washboard. I gave mine to family next table- 3 kids - 2 washboards for them and a few tears over being left out. You will have a good time.
Hoop de doo is a blast. We've always gone as a family, so personally cannot imagine any other way, however, if you are used to going solo, it would be fun too! Its such a fun, happy go lucky kind of night. I'm sure you'd love it.