I feel like you don't really need one. A guitar is only an octave higher and it can still cover a lot of the notes that a bass guitar does. Also, you could just use an eight-string guitar and tune the lowest string down to a low E. I know that some bands have eight string guitarists as well as bassists and the bassists play the same notes as the the eight string guitar players, but the bass will make the low end sound "fuller" because of how thick the strings are. But I don't think that really makes any much of a difference. I know this question might make me appear ignorant, but this is my own judgement.
If a piece of music is arranged for a string quartet, you need 2 violins, a viola, and a cello. Sure, you can play it without the cello, but it won't be the same arrangement.
Likewise, popular songs arranged for standard rock bands require drums, bass, one or two guitars, and maybe some other instruments such as an organ or a sax or something, depending.
So if your music is arranged for a bass part, you need a bass. Don't try to play Iron Maiden or Rush without a bass guitar unless you want to fail hard.
On the other hand, there are a handful of bands out there that did not arrange their music for a bass guitar. White Stripes, for example. The Moaners. Maybe your originals band. Some bands use keyboard or synth bass.
Bottom line: if the arrangement calls for specific instruments, you need those instruments to do it right. Since the majority of popular music, from Top 40 to prog metal, is arranged for bass guitar, it makes sense to have a bassist in your band most of the time. Even the Doors, who didn't officially have a bassist, relying instead on the organ, often used a session bassist on their studio recordings. Riders on the Storm only sounds right if your cover band has a bassist to play that main riff.