You have to make sure your switches have a management VLAN set so you are always sure they will primarily try to get an address on the mgmt VLAN behind the Fortigate. When that fails for any reason the switches will attempt DHCP on any VLAN which can include the VLAN 501. If they go online that way everything should still work though but your switches would be in a different VLAN. If there is something on the second ISP blocking access to dashboard then yes the switches would fail to connect but traffic should still be forwarded. Also since you have a stack, why don't you create aggregates on your Fortigates? It is best practice to also not cross an ISP over two stackmembers. So you can either keep both VLAN 500 and 501 ports (should be six in total) on your left switch or have VLAN 500 on switch 1 and 501 on switch 2. Then if one switch goes down you don't lose both your ISP's.
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