I have a working shiny app which uses library(coral), where coral is a package I wrote myself, installed in the directory of the shiny app.
When I attempt to deploy my
shinyapps.io has the devtools package,
and it can install packages from GitHub,
if you don't mind putting your package there:
library(devtools)
devtools::install_github("youruser/yourpackage")
devtools has other functions too to install from other sources,
for example install_bitbucket() from bitbucket, install_url() from an arbitrary url and install_file() from a local file on disk.
To install from a local directory, you can use install:
devtools::install('path/to/r/package/dir')