If the problem is with a third-party vendor/partner that your company has a solid relationship with, then the best person to ask is the vendor. Get in contact with someone at their company, preferably someone associated with the team building the API, and ask their opinion; they will probably know the ins and outs of their application fairly well, and be able to offer good advice.
On the other hand, if it's a fairly stable, well-designed, preferably RESTful webapp, you should be able to send HTTP requests of various types at it, parse the responses, and have it work.
There's also always the old programmer standby of adding another layer of abstraction. Depending on the nature of your company's relationship with the vendor, you may be able to get the actual interface of the API before it's completed, or even just get a general idea of how it will work. At that point, you can write your own "API" internally to emulate theirs using the webapp's existing features. Once the real API arrives, you should be able to quickly convert your implementation into being nothing more than a thin wrapper around the proper API. Then, hopefully, you can ultimately refactor the wrapper layer out of existence, without ever breaking functionality. It's not a very pretty, or purist solution, but if the vendor is willing and able to work closely with you, or even just willing to throw a few pages of API documentation at you, this might be a fairly practical way to get the functionality working now, while being able to quickly and painlessly (as much as is possible) switch over to the proper API when it becomes available.
In any of these cases, if the webapp is being hosted on servers owned by the vendor, you really do need to clear this with them first, especially if your app is going to be hitting theirs for data with any regularity. If the webapp is instead part of something hosted internally in your company, then you should similarly check with your network/system admins, give them an idea how heavy the traffic will be, and make sure they're okay with it. Failure to alert people before slamming their server with traffic is not cool, and may result in things like being IP banned from accessing them, making their company more hostile to yours, or even getting yourself fired when both those things happen, and management starts looking for someone to blame so they can get back in the vendor's good graces.