I would suggest checking out the R logs on shinyapps.io and see if you get anything different in Chrome vs. Firefox. The other place to look would be the "network" tab in "Firefox Developer Tools." Go to "Menu" -> "Web Developer" -> "Console" or "Network"
That shows you a bit more information about what is going on behind the scenes of the waiting you are stuck in.
Thanks for your help Cole. This pointed us in the right direction.
We ended up finding the problem in the page source. We had a selectizeInput's options parameter set to a vector with a large number of items, included mostly duplicate values. (We had just set the options as a column in the data, missing the unique() call that we used elsewhere). These options were being parsed into thousands of lines of html. We made this vector unique, and firefox works!