There is no way to check history or status. They use a website only for reporting and once your shops are submitted they just go away.
Keep good records yourself. Back before mobile deposits and ATM machines that included a copy of the check deposited on your receipt, I used to photocopy every check from them because they never referenced what the shop was for. These days I do a mobile deposit of their checks and keep the original check in my file. When I can't reconcile an amount, I simply email my scheduler. And from time to time a payment got lost in the cracks because the way their system worked (and seems to still work) is that the scheduler approves your invoice and sends it on to accounts payable. If they forget to approve it or forget to send it to accounts payable or accounts payable has a question they forget to ask, your pay can go to limbo until you shake the bushes a little bit.
Mystery shopping is only one small piece of Harland Clarke's business and it is less systematic and organized than with companies who do nothing but mystery shop. At the same time, I have never found them to back away from a payment due and from time to time I have been paid more than I invoiced for and we either worked it off against future jobs or considered it a 'bonus'. They do tend to pay within 2-3 weeks and if I haven't seen payment after a month, I drop them an email to check on it.