The problem is not a dependency of Emacs 26.3, but a dependency of this specific compiled executable. The compiled executable that you have needs
libMagickWand-6.Q16.so.3, which is provided by the package
libmagickwand-6.q16-3 in Ubuntu 18.04 bionic. Ubuntu 19.04 disco comes with
libmagickwand-6.q16-6, which contains
The number after
.so. in the name of the shared library is the ABI version. The ABI version reflects the exact format of the data structures that the library uses to communicate with the application. If the format of these data structures change, for example to support new functionality, the application needs to be recompiled.
You can install multiple versions of the same library with a different ABI. That's why the library file name contains the ABI version and the Ubuntu package name also contains the ABI version. Ubuntu only lets you install a single version of the ImageMagick application, because they'd all compete for who gets to be
/usr/bin/display-im6 and the like. But you can have both
libmagickwand-6.q16-6 installed. This capability is why shared libraries always come in a separate package from other types of files.
Note that you can search which package contains a file on the Ubuntu website. (There's also a local tool,
apt-file, but it won't let you query versions of the distribution other than the one you have installed.)
If you don't mind using Emacs 26.1 instead of 26.3, use the version provided by Ubuntu. Using the applications provided by your distribution is always the simplest solution.
If you want Emacs 26.3 and you compiled it Emacs yourself, just recompile it with the development packages from your current version of Ubuntu.
If you got Emacs 26.3 from a third party, see if they have compiled binaries for disco.
If you just download
libmagickwand-6.q16-3 and any other library that it may require and install them manually, you'll be able to run your existing Emacs executable. However beware that this way you won't get security updates.
You can add Ubuntu bionic as a package source in apt (make sure to include the security updates from
bionic-security). This way you can install packages from either bionic or disco; when the same package name is present in both, you'll always get the most recent version.