Welcome to the community, @Drew_S!
No, API calls do not count towards your automation runs.
Although there are rate limits:
The API is limited to 5 requests per second per base. If you exceed this rate, you will receive a 429 status code and will need to wait 30 seconds before subsequent requests will succeed.
If you anticipate a higher read volume, we recommend using a caching proxy. This rate limit is the same for all plans and increased limits are not currently available.