Does anyone know if it's possible to convert the AIR-AP1120B from autonomous to LAP?
Sadly, the 1120 with "B" Radio is not supported :(
Support for the 1121 was added in WLC version 126.96.36.199 Have a look at these related docs;
Software release 188.8.131.52 New Features
LWAPP for Cisco Aironet 1100 Series Access Points (802.11g radio only) Allows the Cisco Aironet 1100 Series Access Point to be upgraded from autonomous access point mode to lightweight mode using the autonomous to lightweight mode upgrade tool.
Note You must install software release 184.108.40.206 on the controller before connecting 1100 series access points to the controller.
LWAPP Upgrade Tool Troubleshoot Tips
Access points must run Cisco IOS Release 12.3(7)JA or later before you use the upgrade tool.
You can use the Autonomous to Lightweight Mode upgrade tool to install Cisco IOS Release 12.3(11)JX on these access points:
**All 1100 series access points containing MP21G (802.11g) radios
All 1130, 1230, and 1240 series access points
All modular 1200 series access points running Cisco IOS software and containing these supported radios:
802.11g: MP21G, MP31G
802.11a: AIR-RM21A-x-K9, AIR-RM22A-x-K9
All 1300 series access points in access point mode
Cisco has released a free tool called the "Autonomous to Lightweight Mode Upgrade Tool" that allows selected Cisco Aironet autonomous access point models to be configured for lightweight mode operation.
The Autonomous to Lightweight Mode Upgrade Tool supports the following models:
Cisco Aironet 1240AG Series access points
Cisco Aironet 1230AG Series access points
Cisco Aironet 1200 Series access points that contain 802.11g (AIR-MP21G-x-K9) and/or second-generation 802.11a radios (AIR-RM21A-x-K9 or AIR-RM22A-x-K9)
Cisco Aironet 1130AG Series access points
**Cisco Aironet 1100 Series Access Points that contain 802.11g radios (AIR-AP1121G-x-K9)
Cisco Aironet 1300 Series Access Points/Bridges (AIR-BR1310G-x-K9 or AIR-BR1310G-x-K9-R). A Cisco Aironet 1300 Series operating in Lightweight Access Point Protocol (LWAPP) mode only operates as an access point. This series does not support LWAPP bridging mode.
The Autonomous to Lightweight Mode Upgrade Tool supports a process to migrate an autonomous access point from autonomous mode to lightweight mode. Unlike a VxWorks to Cisco IOS Software upgrade, this process is a Cisco IOS Software upgrade to the existing Cisco IOS Software image-not an operating system "swapout". In converted access points operating in lightweight mode, Cisco IOS Software continues to run on the access point, while LWAPP is used to communicate with a wireless LAN controller. Since LWAPP supports automatic access point configuration, there is no need to retain or convert the original autonomous Cisco IOS Software access point configuration.
Cisco Aironet Access Point Support for Lightweight Access Point Protocol
Hope this helps!