The Access Table shows the entire Bishopric (including clerks) as not having access to Payment Requests. However, that seems to be true and not true. It is true that a Bishop cannot enter expenses from the "Unit Leader/Clerk" screen of the Budget page in LCR. However, if the Bishop switches the drop-down to another selection such as "Elder's Quorum", the "Payment Request" tab appears, and a payment request can be submitted by him for that org. I believe there's other threads on the forum that cover this in a bit more detail, bottom line is that there's a bit more complexity to what is and isn't available, than what the Access Table can always relate.
And I don't know if there's any differences between the access available via LCR and the Tools app, my comment above was specific to the LCR interface. And I also don't know if everyone in the Bishopric/clerks callings has the same ability to switch to another organization and enter a payment request (I'd guess Bishop, counselors, ward clerk, and finance can do that. But just a guess).