I know there are plenty of connecots and API documentation to bring data into Domo. However, is it possible to use the API to pull data from a Domo card to another software?
it is possible to use the API to import and export datasets. The documentation is here and specific export instructions are here, share it with a developer, he or she will understand the details.
Essentially, you send something like
GET https://api.domo.com/v1/datasets/317970a1-6a6e-4f70-8e09-44cf5f34cf44/data?includeHeader=true&fileName=Q4_Sales_Data.csv Accept: text/csv Authorization: bearer <your-valid-oauth-access-token>
and the dataset will be returned to you. This does not afaik work for the data in cards (and things like filters, sorting and beast modes), but it would work for the dataset you use to power a card - in case of doubt, make the dataset as you want using a dataflow and then just request that one.
PS: you might also need to talk to Domo Support to enable the connection on their end.