Does perl have something analogous to the shell's source?

Not for shell code, as shell code needs to be interpreted by a shell (likewise, a shell cannot run Perl code).

As for "the shell which the Perl script is running in", Perl is started from a shell, but it's a separate subprocess, so strictyl speaking, the Perl script is not running "in" a shell.  This means the same child-parent restrictions apply, i.e. the Perl script (nor any further child processes started from it) cannot set the environment of its parent shell.

Update: maybe this clarifies it somewhat. When you run something from the command line, for example

$ perl -e 'system "ps Tf ; sleep 1"'
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you see child-parent relationships as follows (indenting means "child of")

  PID TTY      STAT   TIME COMMAND
29353 pts/7    S      0:00 bash                                   # th
+e initial interactive command shell
29388 pts/7    S+     0:00  \_ perl -e system "ps Tf ; sleep 1"   # yo
+ur Perl script
29389 pts/7    S+     0:00      \_ sh -c ps Tf ; sleep 1          # a 
+shell started by system()
29390 pts/7    R+     0:00          \_ ps Tf                      # a 
+subprocess run by the shell
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All those are separate processes created via fork/exec, and no child is able to manipulate the environment of its respective parent process. But it does inherit the environment of its parent.

Does perl have something analogous to the shell's source?

I'd like to apply the initialization scripts to the shell which the Perl script is running in.

I'd like to apply the initialization scripts to the shell which the Perl script is running in.

As for setting an environment variable inside a program or script and have that change affect the parent shell, that's a FAQ, both a Unix FAQ and a Perl one (see perlfaq8). See also: export environment variable from a perl program.