The API they provide is very good: https://developer.tradier.com/documentation
If you call/email them, they will give you your personal key, so that you do not need to worry about oath 2.0 authentication, which would be necessary if you developed software for somebody else. It is an important point, and will save a lot of time.
Then you can use their /positions and /quotes API to get what you currently have open and the current prices (both underlying and options).
Then you will need to use some heuristics to combine the legs into positions (date/time mostly). If the straddles are rolled, it gets complicated, as the positions need to be combined differently and the cost basis changes. But even without those complications with rolls, the information you get is already good enough and not less helpful than what, for example, TOS mobile provides.