Many species of felines purr, producing both sound and tactile vibration. With suitable electronics and small speakers, we can reproduce the sound of purring.
purring.org has lots of information about purring, including audio recordings of domestic cats, cheetahs, and servals. Based on this information, we can predict what other feline species would sound like and synthesize our own purr audio.
Existing off-the-shelf boards, including the Robertsonics Tsunami, can read audio files from a memory card and play them. The Tsunami has many options for controlling the playback of sounds, and can change their volume over time. Because it supports up to 2048 audio files and can play several sounds simultaneously, it could also be used for sound effects beyond just purring.
Standard small speakers should be a reasonable way to make the purring audible.