In the US, "pull in" a date and "push out" a date are common. You will get some hits with a google search. google "pull in a date" google "push out a date" citation: https://english.stackexchange.com/questions/5133/how-to-say-i-will-try-to-move-it-to-an-earlier-time-or-what-is-the-opposite-o We typically say "push out" dates to mean a delay in date or time and we "pull in" dates to mean advance or "prepone" a meeting. "prepone" is meaningless in English outside of India."push out" and "pull in" are accepted terms when speaking to schedulers who use any tool that creates Gantt Charts.The tool automatically shows an item being pushed out to the right or pulled in to the left if you change the due dates.

I would say: we can advance the shipping date to..... in this context, because it's a commercial background. TBH I have never even heard pull in in this context.