A rainbow is not a physical object that has a position. It is an optical phenomena that depends on your location relative to the sun and rain. If you are standing where your eyes can intercept the colored light, you are standing with your back to the sun and the sunlight is reflecting on raindrops in front of you. Someone else standing in a different location would not necessarily see a rainbow if they looked up at the same part of the sky.
From University of Illinois:
According to Descartes' calculations using laws of optics, the three stage refraction-reflection-refraction pattern that light undergoes when passing through a raindrop produces a concentration of outgoing rays along a line that is 42 degrees above the head of an observer's shadow. This concentration of light rays is the rainbow that we see.
Also this National Geographic article has a nice description:
Viewers on the ground can only see the light reflected by raindrops above the horizon. Because each person's horizon is a little different, no one actually sees a full rainbow from the ground. In fact, no one sees the same rainbow—each person has a different antisolar point, each person has a different horizon. Someone who appears below or near the "end" of a rainbow to one viewer will see another rainbow, extending from his or her own horizon.
What this means is that a rainbow is not really a meteorological occurrence that can be measured or catalogued, because you would get a different answer depending on your reference point. Lightning, in contrast, is a physical phenomena that has a precise location which can be determined and verified from multiple points of reference.
Rainbows are photographed and archived by enthusiasts, but it's really about artistic appreciation. While I have not looked into rainbow forecast services, a quick search shows some interesting resources, such as How to Predict Rainbows and Plan Photographs which has a link to a photography app that can plan for rainbows and here is a rainbow app you can install on your phone.
How Rainbows Happen also has a nice description (shown below) and some other useful resources.