Making my own radio transmitter
That would work, as the comments correctly noticed: This is a software defined radio.
However, unless you plan to generate IF with your sound card (which you most likely aren't doing, since sound cards, even ones that support 192kHz sampling rate, aren't frequency flat for a large range), you will only be able to generate carrier-symmetrical spectra with a real-valued baseband signal (for the ham in you: no SSB!); what you want is to be able to mix up one signal with a cosine @ f_carrier, and one with a sine @ f_carrier (notice the phase shift); that would be called IQ mixing. You can actually do that, and there are commercial, non-commercial and half-commercial solutions out there that implement that for you.

Now, a sound card's bandwidth is not really large, so as soon as you start investing yourself much in this, you might find it unsatisfactory, but for a start, what you're planning is definitely possible.