Before the client could invoice for an amount including GST they would have to be registered with CRA. If the client is not registered then they cannot bill the company nor can your client claim GST on this amount.
Does your client claim this amount as revenue in her business? If not, then there could be payroll issues for the company and your client.

My client is registered for GST and she charges it on her regular customers. However, for this one they just pay her a flat amount each month and they don't add GST. Since she does not invoice them she can't add GST to an invoice. It is claimed as revenue on the books. I just don't know if stipends are subject to GST. If they are then she either has to go back to them and ask them to add GST to the flat amount or she has to absorb the GST as part of their payment (eg. $700 would be $666.67 revenue and $33.33 GST).

If your client is registered for GST then this payment should attract GST. However, if there is no invoice showing the GST and with her GST number then her client cannot pay the GST. The other alternative would be to have a contract between your client and her customer where her payment is referred to and whether it is GST included or to be added. The contract will not require a regular invoice.
Stipends are payments for services rendered and would attract GST.