The RPi has one built-in serial port but in it's default mode, it only supports TX / RX and GND pins. That won't help you for PTT. Now, what you can do is invoke one of the RPi's alternative I/O mappings and at that point, you have full access to the flow control pins (specifically RTS) that most people use for PTT. Please note, the RPi is a 3.3v device with limited current so you'll probably need a level shifter to drive the PTT line on your radio.
Ps. In addition to considering Soundmodem, have you heard of Direwolf? This soundcard TNC supports is being actively developed, supports multiple parallel decoders for off frequency signals, supports multi-bit correction, etc. None of those times is true of Soundmodem.