So you'll need to use a custom buildpack to achieve this which will grab wget, compile it and then include the resultant binary into your slug (a heroku term).
Turns out, I've just whipped one up - https://github.com/johnbeynon/heroku-buildpack-wget.
To use;
create a .buildpacks file in the root of your project containing
git://github.com/heroku/heroku-buildpack-ruby.git
git://github.com/johnbeynon/heroku-buildpack-wget.git
and then do
heroku config:add BUILDPACK_URL=git://github.com/ddollar/heroku-buildpack-multi.git
Now, when you deploy your application, it will use the heroku-buildpack-multi which will read your .buildpacks file and use those defined in there. heroku-buildpack-ruby is the default provided ruby buildpack and then mine will add wget into your application.
to verify if it's work, do heroku run bash and then try and run wget and see if it works.
/tmpfolder. I'm serving some content to the user. - Mika H.