You don't need a FortiGate to manage the FortiAPs, you also have the option of doing it via the cloud.
Goto www.forticloud.com and signup for a free account. Next follow the steps in the FAQ here on how-to join the AP to FortiCloud.
The main bits that may be of interest to you are:
How can I register a FortiAP to my FortiCloud account?
Supported FortiAP models include a sticker with a unique FortiCloud key affixed. This key must be entered into the FortiCloud interface to register the FortiAP to your FortiCloud account
I have an older FortiAP that doesn’t include a FortiCloud key. Is there some way I can add my device to a FortiCloud AP Network?
Older FortiAPs that have shipped without a FortiCloud key can be added to FortiCloud. Open the FortiAP management interface, and in WTP-Configuration select FortiCloud. Enter your FortiCloud credentials, and select Apply. Login to FortiCloud and select Inventory > Unused APs to see the list of FortiAPs. Select Deploy to AP Network > Existing AP Network.
Is there a minimum firmware version that I need to run on a FortiAP for the FortiCloud AP Network feature to work?
The FortiAP must be running FortiAP OS 5.2 at a minimum