Does Windows have a built-in ZIP command for the command line?

It’s not built into Windows, but it’s in the Resource Kit Tools as COMPRESS,

C:\>compress /?

Syntax:

COMPRESS [-R] [-D] [-S] [ -Z | -ZX ] Source Destination
COMPRESS -R [-D] [-S] [ -Z | -ZX ] Source [Destination]

Description:
Compresses one or more files.

Parameter List:
-R Rename compressed files.

-D Update compressed files only if out of date.

-S Suppress copyright information.

-ZX LZX compression. This is default compression.

-Z MS-ZIP compression.

Source Source file specification. Wildcards may be
used.

Destination Destination file | path specification.
Destination may be a directory. If Source is
multiple files and -r is not specified,
Destination must be a directory.

Examples:

COMPRESS temp.txt compressed.txt
COMPRESS -R *.*
COMPRESS -R *.exe *.dll compressed_dir