You don't need to know many chords to play songs on ukulele. Here are some easy song recommmendations.
Once you learn the beginner ukulele chords (Am, C, F, G), you can play a ton of four chord songs. I made giant list of four chord songs ordered by chord progression. You may have to transpose the songs.
Here are some other song recommendations that don't use many chords.
You don't have to start with 4 chord songs. You can play most songs on the ukulele, even Through the Fire and Flames. If there's a song you like, try Googling the song chords, and give it a shot! The first song I learned on ukulele was I Will Follow You Into the Dark. I played along to this cover many, many times before making my very first cover. Don't be intimidated by hard chords. Transposing the song often simplifies chords significantly, and you'll get it with practice. Here are more tips on understanding song chords.
I recommend keeping a song collection of songs you play. I started out using a bookmarks folder of song chords links. When this got unwieldy, I moved to using Google docs. Eventually, I moved my song collection to MusicParsed to solve the following problems with other chord sites: interruptive scrolling, inaccurate chords, and annoying pop-ups.