I'll also answer the implied question, which is: "How do I figure out how to play a song from the recording?"
The best way to figure out how to play a song is, for me, to attack it piecemeal. Figure out the easy bits first, then move along to the harder stuff.
The general idea
My procedure is to first write down the lyrics, leaving room for chord changes. (If the piece has no lyrics, I'll sometimes make a list of the different parts of the song, just as a starting point.
Next, I figure out the easiest parts of the song. This is generally the verses or the chorus, but it can be anything. Whatever parts of the song you've figured out, write those chord changes down above the lyrics. If the song has a main riff, work on that as well.
Tricks that work for me
I usually find it easiest to find bass notes. From there, it's easy to figure out if the chords are major, minor, seventh chords, or whatever. (You can try playing both major and minor and seeing what sounds closer. If needed, refine to seventh chords, minor ninth, whatever. This is where experience will build up an instinct for this.)
Also try humming a note along with the recording, pause the recording while holding the note, then find it on your instrument. (Stringed instruments make this easier, since you can slide up on the neck until the note is reached, I have a harder time figuring songs out on the piano than the guitar.)
Chords and the song's basic key
You say there's no instrument playing chords? Untrue, there's harmony in almost every song, and there will be implied chords if nothing else.
You should now know what key the song is in, which will make figuring it out much easier. Keep figuring out various parts of the song until you have a complete lead sheet in front of you. If there are tricky parts, you may want to write them down in musical notation or tabulature, whichever works for you.
How do you know what key the song is in? Often, the first chord in a basic verse-chorus song is the key the song is in. Again, use trial and error to whittle the key possibilities down to a few major and minor keys.
Once you develop your transcribing skills, you'll start finding mistakes in online transcriptions. Please also note that many songs have chord sheets and tabulature on the internet, sometimes even on the artist's own site. These can give you a leg up and save you a lot of time, but they can also be oversimplified, or alternate versions of the song, or just plain wrong.