A compiler is required only to support the instruction set of the processor. Board specific support itself is not the responsibility of the compiler. Moreover in the case of ARM based devices, only the ARM core is the responsibility of the compiler; support for peripherals, and in particular the PLL clocking scheme for your part are also not the responsibility of the compiler. So the fact that your board is not mentioned in the documentation is largely irrelevant.
Some tool-chains and particularly IDE's from the likes of Keil (an ARM subsidiary) and IAR provide a huge amount of chip and board support. Both these companies provide demo versions, Keil's I believe is good for at least 32K, and the IDE also supports free GNU tools such as Code Sourcery, which has no restrictions on code size.
However in many cases and in particular the case of the STM32, a lot of the board support you need is available from the chip vendor. ST provide a standard peripheral library and ARM CMSIS (ARM's standardised Cortex-M support API), plus many other libraries besides. Keil actually ships ST's Standard Peripheral Library and CMSIS with its toolchain, but getting it form ST ensures the latest version. The libraries and manuals can be found ion the Design Support tab here. The board page linked in your question has a link to download example code to support the off-chip board peripherals.
So the upshot is that Code Sourcery or other ARM GCC distribution will work fine, but you may need to build your complete development environment from parts such as chip or board vendor supplied libraries plus an IDE/debugger environment such as Eclipse or uVision.
With respect to using Eclipse, there is an article here describing how to do it for the STM32 Discovery board; most of which is applicable to your board at least for building if not for debugging/flash programming - the Discovery board has a built-in STLink JTAG debugger.
Update June 2021
A lot of the above is out-of date ST have largely deprecated support for the SPL and do not even provide it for newer series such as STM32F7, replacing it with he STM32Cube framework.
Importantly however ST now provide a free IDE with a toolchain and CubeMX integrated: https://www.st.com/en/development-tools/stm32cubeide.html