Will libhandy be ported on Ubuntu 18.04?

I suggest you report a bug to show there is interest in the library for the latest LTS.

Manfred Hampl suggests this article as an answer to your question:
FAQ #3037: “no rolling release”.

The information in that FAQ is also applicable for backporting packages that do not exist in older releases.

What you could try is downloading the libhandy-0.0-0 package for disco and manually installing it on your system. A quick check let me assume that all dependencies can be met on bionic.

I wrote an application that depends on the library. I know of other applications who depend on it, too.
If the library is not ported, there is no way the application will work out of the box in the LTS, right?

As long as the libhandy-* packaged do not exist in the repositories for Ubuntu 18.04, any application that depends on it will not work "out of the box".

Suggestions:
1. Create a PPA and try compiling libhandy on Bionic as a proof that it can be backported. If that works, then adding your PPA to the list of repositories will enable everybody to use your application.
2. create a bug report in bionic-backports to backport the version form disco to bionic (eventually the same for cosmic).

Have you tried manually installing the version for disco? Something like
wget https://launchpad.net/ubuntu/+archive/primary/+files/libhandy-0.0-0_0.0.7-1_amd64.deb
sudo dpkg --install libhandy-0.0-0_0.0.7-1_amd64.deb
(and eventually the same for gir1.2-handy-0.0 libhandy-0.0-0-dbgsym libhandy-0.0-dev libhandy-0.0-dev-dbgsym)

Looking for libhandy-dev for 18.04 i386 to compile some budgie applets.

@David Robert Lewis:
The name of the package is libhandy-0.0-dev and it is available e.g. in ppa:ubuntubudgie-dev/budgie-extras-daily