There are a lot of places selling "organic acids tests" that include 5-HIAA (along with plenty of other interesting analytes). However, from what I can tell, they're all just rebrandings of the products from these four sources:
Metametrix / Genova Organix (the two companies have merged under the Genova name). This is a single urine sample test. True Health Labs is the cheapest provider I've found, but it's still $329.
Great Plains Labs Organic Acids. This is also a single urine sample test. MyMedLab is the cheapest provider I've found at $325. You can order directly from Great Plains, but they require a healthcare provider's signature with the returned kit.
Doctor's Data Neuro-Biogenic Amines, Comprehensive. I can't tell whether this is meant to be a single or 24-hour urine sample. The cheapest provider I've found is Integrative Psychiatry at $349.
ZRT Labs NeuroAdvanced. This is a dried-urine spot test. The cheapest source I've found is ordering on Amazon for $330.
Options when going through a healthcare provider
If your doctor is friendly towards ordering tests:
A 24-hour 5-HIAA test is CPT code #83497, and is offered by both Quest and LabCorp.
If you need to find a doctor friendly towards ordering tests:
Mary Shomon has a directory of doctors that are familiar with thyroid issues. I suspect such doctors are often more likely to be friendly towards people that want to try test-driven approaches to their health.