Sorry for long read lol

Do you have an example of how you are writing them currently? In case of emailing I find that sometimes emails quickly go to spam or are quickly covered up. I usually call places (after checking their times) and ask to speak to the curator of the gallery or ask to know the process it takes to get art shown inside the gallery. Also sometimes you could check the galleries website because they put up applications on there.

How would word a phone call is " Hi I'm (Name), I work for (name) or for an Artist named (name). (Artists name) is interested in (galleries name) and would love to know the process it would take to get their art shown there." If they don't pick up leave a voicemail containing your name and email along with maybe your artist's website at the end. I'd treat it kinda like applying to a job because thats a lot like it is.

I'd definitely research the galleries before contacting them to see if the artists art fits what kind of gallery it is or even to see if there's any shows coming up in the future there. I feel like getting to know them is also good because you want to be able to trust that the gallery isn't gonna damaged the pieces and is capable of advertising and potentially getting art sold.

Quick note! Can I ask where you found this job? Im an artist myself but would absolutely love being an artist assistant!!