Take a look at the Bridge Library Process Tutorial.
That example shows using the curl command to retrieve an ASCII text file. You will need to change it to use the required command for your API service.
For example, something like:
Process p;
p.begin("curl");
p.addParameter("-d");
p.addParameter("\"devid=v0123456789ABCDE\"");
p.addParameter("http://api.pushingbox.com/pushingbox");
p.run();
I did not try to compile or run that, so you may have to make some changes to get it to work, but it's a starting point. Also, as mentioned in the tutorial, you will need to include process.h and call Bridge.begin().
Note that the parameter that includes the devid string includes quote characters inside the string, which are escaped with a backslash character. When updating the devid string, be careful not to delete the backslash that is escaping the ending quote character.