First of all, there is little need to worry about trademark infringement. Trademarks are only protected when used in commerce If the customer is only ordering one of the item, it almost surely is not being sold commercially. So unless sit is or might plausibly be used to advertise goods or services, any use of a trademark is not trademark infringement.
As for a possibly copyrighted image, it is harder for a printer to tell. The customer might have permission, or the reproduction might qualify as fair use. That will depend on how the image was licensed, or how the customer plans to use it, and the printer can't know either one.
One possible approach is to require every customer to sign a statement that any copyrighted content is being used with permission, or under a valid claim of fair use. That will help protect the printer, although it might not help in a blatant and obvious case, where no reasonable person could have accepted it as valid.
The safest approach is to simply refuse all orders where it seems likely that copyrighted content is being reproduced. However, the vast majority of such orders would not be cases of infringement, so significant business might be lost that need not be.
One should also remember that there is a penalty for copyright infringement only if the copyright holder notices and decides to sue. If a single item is being made for private use, what is the chance of that? Probably fairly small.
Over all, this is something that the printing company should probably review with its lawyer, to decide what policy reduces its risk to a level that it can tolerate. Such a policy decision is beyond the scope of this forum.