[lucaslazarus]

Looks to me like initialisms get periods unless they are acronyms or trademarks. So "ICE" is fine because it is read as "ice" and not I.C.E. (eye-see-ee). REI is not an initialism though, but presumably they have kept it as-is because it's their trademark/"doing business as" style.

[hopelite]

That is correct and also how everyone else should be using the English language.

The “REI” initialism, is the official d.b.a. for the Recreational Equipment, Inc. company.

[lucaslazarus]

Erratum: I meant to say that REI is an initialism but not an acronym