I'm not very knowledgeable on Helicobacter pylori issues, but I thought it was cool that there's some options for testing it cheaply. So here my notes on sources for ordering the testing yourself. Please leave a comment if there's any notes I should add.
These tests are meant to show whether you've developed H. pylori antibodies, so they might show you a positive test result even if you're no longer infected. The breath and stool sections below have more expensive tests that test for active infections.
I haven't found any information of how accurate these test kits are, but a similar-looking medical supply option says it has 90% sensitivity and 78% specificity.
Here's a video of the steps involved with performing the SELFCheck brand test.
Blood testing looks for antibodies like the at-home tests do, so it could share the downside of giving a positive result when you're no longer infected. I suspect it's more reliable than the at-home tests though.
You'll typically get a collection kit in the mail or picked up from a local office, and then either mail it back or redeliver it to the local office depending on which company you're working with.
For this test you give a breath sample before and after consuming a special capsule or drink that contains urea. It looks like you typically get results back in 3-4 business days.
I only found one consumer-ordered testing option in Australia:
You can also see the international blood spot test information below.
All of these UK-based companies ship the blood spot tests internationally: