I think you'll want to build ImageMagick so that it's statically built. For example:
./configure --disable-shared --enable-delegate-build
Doing this will make your version portable and completely independent of any system libraries.
Follow up questions
If I have webp installed, will this include the library in the installation?
Probably not. It won't include any external dependencies such as webp or ghostscript (gv). There isn't any way to include them either as a function of ImageMagick. You're options are to lean on the package manager and include them as "dependencies" or roll them into a tarball yourself and deploy them as binary blobs.
Suppose I have the webp binaries (either I package them or build them
from source). How do I tell imagemagick where they are, assuming I
build with the --with-webp option?
If they aren't in a standard location,
/lib for example, and they aren't hardcoded into the executables via
config then you can use the environment variable
$LD_LIBRARY_PATH to specify where libraries are to running apps.