Include a file dependencies.R with your application or document and include library(dependency) for each package dependency. That file is not loaded by your content, but its presence in the deployment is enough for rsconnect+packrat to list those packages as required.
The packrat package is responsible for identifying dependencies within your content; it does not know about the import package or how it adds packages.
It looks like someone recently filed an issue against packrat asking for import support: https://github.com/rstudio/packrat/issues/527
The internal packrat function identifyPackagesUsed is where most code scanning occurs. It would need to know about the ways that import adds package dependencies. https://github.com/rstudio/packrat/blob/505f12aa4ef884cc7021719f4f0e4b69a0b93354/R/dependencies.R#L401
Until the reference packrat issue is solved, the dependencies.R option is the most straightforward way to get your dependencies correctly enumerated. Unfortunately, it means declaring those references in two places.
CC @kevinushey