If a company wants to support a charity, let it donate its OWN funds. Otherwise, its a crass PR game funded by customers. Gross on every level.
People rarely trust stores, so Im not even sure how that relates in your question. Its a store. Its useful. So is the next store that will replace it. So not only is there no special trust in a store, but even if there was that would never carry over to some org theyre fundraising for.
I make exceptions to give to a select few local charities that might have a donation box at the register, with no ask by the cashier. For example, our local pet food store had a collection can for our local Trap/Neuter/Release program, which I would give to anyway, but having the can there was a convenient way to give a bit extra. I give a pretty hefty amount to a select group of charities, by the way, through direct volunteer service, in-kind donations, and in certain circumstances, significant cash. So, Im anything but anti-giving.
But there are more reasons for saying no:
1) The pushy store campaigns always give to a small number of safe and easy charities. Its always the largest nationwide charities and about kids, kids, kids, whether its a kids medical charity, or some bollocks like Make A Wish. These are the easiest categories of charities to fundraise for, regardless. Stores dont want to be associated with truly useful nonprofit that isnt as easy to fundraise for because of peoples attitudes, that might make someone upset, whether Planned Parenthood, or a charity helping women out of prostitution, supporting LGBT rights, or a wildlife rehabilitation or conservation organization with local impact. If I ever encounter an Arbys asking for donations to The Lilith Fund, after I pull myself off the floor, Ill certainly donate.
2) Its all about public shaming into giving, like Children International and other street charity muggers. I will not play that game. Ok, I lie. I do play that game. I play the game of wrecking their street corner for them if theyre truly obnoxious. One particularly memorable occasion was the day I was accosted by a charity mugger for a generically-named environmental org with no credentials for actual work beyond collecting cash, when I was going home after doing a 2 week straight stint of conservation and wildlife education programs with legit fundraising (no pressure, 100% went to the org) on the side for a nationally respected rehabilitation and conservation nonprofit. No was apparently not an ok answer, and the jerk started screaming down the street that I dont care about wildlife. So I turned around and did an educational talk to the passers by about the meager percentage of his take going to benefit wildlife, and how if they want to give for wildlife, they should go directly to a researched and trusted orgs website, not waste $0.92 of every dollar they give to support paid street or phone canvassing groups. He moved on.