Yes, the solution here is similar to the same question for Travis-CI. 
Here's an example of an appveyor.yml file that will enable you to use packrat packages in your package:
# DO NOT CHANGE the "init" and "install" sections below
# Download script file from GitHub
init:
  ps: |
        $ErrorActionPreference = "Stop"
        Invoke-WebRequest http://raw.github.com/krlmlr/r-appveyor/master/scripts/appveyor-tool.ps1 -OutFile "..\appveyor-tool.ps1"
        Import-Module '..\appveyor-tool.ps1'
install:
  ps: Bootstrap
# Adapt as necessary starting from here
environment:
  global:
   WARNINGS_ARE_ERRORS: 0
   USE_RTOOLS: true
build_script:
  - R -e  "0" --args --bootstrap-packrat
test_script:
  - travis-tool.sh run_tests
on_failure:
  - 7z a failure.zip *.Rcheck\*
  - appveyor PushArtifact failure.zip
artifacts:
  - path: '*.Rcheck\**\*.log'
    name: Logs
  - path: '*.Rcheck\**\*.out'
    name: Logs
  - path: '*.Rcheck\**\*.fail'
    name: Logs
  - path: '*.Rcheck\**\*.Rout'
    name: Logs
  - path: '\*_*.tar.gz'
    name: Bits
  - path: '\*_*.zip'
    name: Bits
The important parts that differ from the template are:
 environment:
      global:
       WARNINGS_ARE_ERRORS: 0
       USE_RTOOLS: true
    build_script:
      - R -e  "0" --args --bootstrap-packrat
This enables Rtools for the build, and loads the packrat packages.
It's also important to note that we are excluding - travis-tool.sh install_deps because that would cause the packages you depend on to be downloaded from CRAN, rather than built from your packrat directory. 
Here's an example of an appveyor build for a simple R package where this is working: https://ci.appveyor.com/project/benmarwick/javaonappveyortest/build/1.0.21