乗りに乗る is a common phrase. The verb 乗る itself can be like “to go with [a decision, the flow, something]”. Just as a pedagogical overly literal translation you can imagine it like “get on board with [a decision, a situation, something]” although the usage is a bit different in practice. It can be used when someone gets into a situation or enthusiastically goes along with something. The nuance of 乗りhere is often rendered in katakana (ノリ) and can be used on its own and the word ノリノリ is also related. It refers to a specifically positive situation or atmosphere, especially one that has some kind of momentum. In my experience it is also something ephemeral. You can think of it as being similar but different to 調子, referring to a “rhythm” of something or a situation. 乗っかる just means 乗る but is more informal. So putting 乗りに乗る together it means something like “to get into the swing of things”, etc. In translation it will be contextual but hopefully you can see the general idea.
So if I were to translate this with no context,
本人が乗りに乗って 調べてくれるから、それに乗っかる ことにした。
Since she herself is [in the mood/into it/on a high/whatever contextual nuance] and is gonna look into it for me, I decided to [just go along with it/get on board with it/go with it/whatever contextual nuance].
Hope that is useful.