Yes. Much the same way people interface GSM (cellular) modules to their projects you could go out and purchase a satellite phone and interface it. It will be pricey though.
... and there are a HUGE range of embedded satellite modems available (hint google those three words =).
Of course, there will be a number of design challenges and limitations imposed - not the least of which will be the cost, as mentioned, but it is not necessary to launch your own satellite to use one for data transmission.