Since the value of \thickmuskip -- the parameter which governs the amount of whitespace that's inserted between adjacent math atoms of type math-ord and math-rel -- is 5mu in all document classes I'm familiar with, and since \! inserts -3mu of (negative) whitespace, you may achieve your journal's formatting objective by issuing the instruction
\everymath{\thickmuskip=2mu minus 2mu}
in the preamble. Note that the value of \thickmuskip that's employed in displayed equations is not affected by this directive. (Aside: If the amsmath package is loaded, then you will need to reset the value of \thickmuskip to its default value explicitly for displayed equations, along the lines shown in @egreg's answer.)
I can't help but remark that the journal's requirement to reduce the value of \thickmuskip while not also requiring a reduction in the value of \medmuskip -- the parameter which governs the amount of whitespace that's inserted between adjacent math atoms of type math-ord and math-bin (such as + and -) -- verges on being bizarre or preposterous. Take, say, the equation
1+1=2
The default value of \medmuskip is 4mu. I think it's a very bad idea to reduce \thickmuskip to 2mu while leaving \medmuskip unchanged. I would therefore suggest that you also execute something like
\everymath{\medmuskip=1.5mu minus 1.5mu}
in the preamble
A full MWE:

\documentclass{article}
\begin{document}
$1+1=2$ --- normal look
\everymath{\thickmuskip=2mu minus 2mu}
$1+1=2$ --- bizarre look
\everymath{\medmuskip=1.5mu minus 1.5mu\thickmuskip=2mu minus 2mu}
$1+1=2$ --- cramped (but not bizarre) look
\end{document}
\thickmuskip1.75mu– Fran Jul 09 '19 at 10:08