Managing third-party integrations with your web app

I'm working on a product that has ALOT of competition and most of my customers want to see integrations with other project management software and time-keeping systems. Tapping IH knowledge - what's the best way to manage multiple integrations with your SaaS app? Is there software that can help manage this (thinking zapier, but would need to be able to customize more carefully)?

My guess is that it's going to end up like the following:

1. Make a call to third-party API
2. Map values to a standardized table/database
3. Manage mappings and API calls for EACH service going forward
4. Push and pull continuously to keep things synced

Any suggestions for solo developers to keep this manageable?