Hi there @Ashley43!
You can absolutely apply to become a third party payment provider - the first step in this process would be to become a Shopify Partner, are you currently part of our Partner programme? The minimum requirement which must be met in order to apply include:
For full requirements and definitions of all the terms above, you can read the requirements and all other information in full here. An outline of what the steps for application of your app can be found via this link, this image also gives a brief breakdown of what the process looks like:
I hope this helps, best of luck with your payment provider!