What do kids say instead of "videotape”?

As others wrote, I think "video" is what you're looking for.

Speaking only from personal experience, "video" is in common use in the U.S., at least among the young (say, 2-30 years old), who know little about videotaping and never use the term. "Did you video it?" is common and clearly understood, as is "I videoed it" and "We'll video Christmas morning."

Thankfully, the monstrous word "videoing" has not seemed to gain similar currency. Few would say, "Please move; I'm videoing," instead saying, "Please move; I'm shooting a video."

This is, again, anecdotal, but is drawn from interactions with teens and young adults on east and west coasts as well as Texas.

In my shop we shoot digital video with a camera; more generically (embracing other methods such as screen-grabbing motion footage^†), we capture it. We also speak of shooting or capturing the subject.

Video which is constructed from scratch in software by manipulating digital still images is built and rendered.

^† Yeah, it's still "footage" even when it's measured in bytes, because taken at the level where people deal with it rather than software it's still a linear sequence of "frames".

In casual conversation, I have heard the word video used in the manner which you are describing and I think this is what you are looking for.

Google definition:

verb: video; 3rd person present: videos; past tense: videoed; past participle: videoed; gerund or present participle: videoing

1. record on videotape. "he declined an invitation to be videoed"

I hope this helps!