For most singer/songwriters, there is a definite advantage to being able to play your own songs…not only at open mics, but for friends, for industry people, at gigs when your guitarist cancels last minute. It is not a must and you can definitely work around it by hiring players and or singing to tracks. However if you are a career singer/songwriter, just do it. Start learning.
By learning an instrument, only good things come out of it. You become a better musician. You have a better understanding of the musical framework of your song. You learn coordinate your hands while singing. You can be more in tune with the expression of the song and mirroring the dynamics or tempo with your voice. You become a better communicating to your player what you would like (if you need them still). You become more versatile and honestly more respected from other musicians. Being about to play another instrument is not just about playing for yourself. Once you are decent on your instrument, you could possibly get hired to play for others. Doing that you would meet new people and get more experience.
All good things. Just do it! Take lessons, trade skills with someone, or get some internet help.
Good luck!
Jocelyn