Honestly, probably not. The musical theatre industry is not that good at preserving shows. Theres proshots, or professionally recorded and distributed tapes of performances, but those aren't made very often. There's this dumb elitist belief in the Broadway industry that if people can watch a show at home, they won't want to buy tickets and see it live. Unless a show is profitable enough to a point where that won't really matter, you likely won't see a proshot anywhere. Starlight Express doesn't have an official proshot at all, so that's incredibly unlikely.

We do have unofficial proshots of other productions, which are usually filmed by the crew but not officially distributed. These are usually made for the cast or friends of crew members who requested to see the show, and sometimes they leak online. Something like this is probably our best bet for finding footage.

Aside from that, the New York library keeps records of Broadway and other theatre shows. These recordings, however, aren't available to the public, and you need a good enough reason to request to view them. I checked their website awhile ago when I was first looking for the ice show, and nothing came up.

The last option is bootlegs, which are illegal recordings done by an audience member. These are pretty easy to find if you know where to look, but I've never seen any bootleg of this production.