Many lyricists who don't play an instrument are still able to write killer songs, including my late, great mentor Doc Pomus ("Save the Last Dance For Me," "This Magic Moment," "Viva Las Vegas").
To achieve success they usually take one of these paths (or both!): a) befriend people who are strong where they're weak, in this case by seeking out great melody/chord types, and/or b) develop their own musical skills.
If you want to follow the first of these paths, an excellent first step is to polish up at least 3 lyrics to a state of "undeniability." Once you have some serious BRILLIANCE to present to potential co-writers, you'll see that hungry "ka-ching!" look in their eyes.
A talented songwriter-musician can read through a killer lyric and "hear" the music in their head. You won't have to beg at that point — they'll want to work together.
Obviously writing an "undeniable" lyric isn't an "overnight success" type of thing. Maybe I'm slow, but it took me 7 years of writing, rewriting, studying songwriting, voice & guitar, gigging at local holes-in-the-wall, and generally absorbing knowledge like a sponge before I had ANY success — unless you count winning quarter-finalist in a songwriting contest ($25, which is 1978 dollars is now worth $1000).
So be prepared to "pay your dues" if you're going to commit to this path. Luckily learning to express yourself as a lyricist is the most enjoyable type of dues to pay.
DON'T SELL YOURSELF SHORT... EVEN A CAT CAN PLAY PIANO!
Learning to play the guitar, piano, keyboard or ukulele is a great idea, because even if you never achieve mastery you'll still develop the vocabulary and rudimentary skills to interact on a much higher level with your co-writers.
I taught guitar for many years, and was able to induce even a total newbie to play a 3-chord song in a couple hours. If you learn even 5 chords you can play a huge percentage of the songs on the planet. Learn 10 and the world is your oyster!
It can seem daunting at first, especially if you had negative experiences with singing or playing an instrument in your youth, as so many of us have. (Ugh.) But with a good teacher you can surmount that obstacle and start rapidly expanding your possibilities.
These days there are TONS of resources available online, so it's no problem if you live in the middle of nowhere. I say go for it!