Unless you convinced an FPGA vendor to build their tools for ARM, or used an emulation layer to run the x86 synthesis tools on ARM, you're out of luck. The former is unlikely and the latter would take all day to synthesise even a simple design. You could probably run
GHDL and maybe
GTKWave on a Raspberry Pi, so you could do simulation work, but not synthesis.
However, there is nothing stopping you synthesising several designs on a PC and then loading the resulting bunch of bitstreams to a Raspberry Pi. The Pi could then load them into the FPGA over USB when needed, using
FPGALink. You would not even have to switch SD cards to get a new design. You could even install the Pi+FPGA in a remote location and load new designs into the FPGA over the Internet.
This board I designed recently doesn't fit the usual definition of "shield" (i.e different form-factor, not "pluggable"), but it's cheap, open-source hardware and known to be compatible with the Raspberry Pi:
MakeStuff LX9 PCBMakeStuff LX9 blinkyMakeStuff LX9 speed testIt has a
Xilinx XC6SLX9 FPGA, 16MiB of SDRAM, a 1Mib EEPROM, an SD-card slot and about 50 general-purpose I/Os. You can access the FPGA's JTAG for programming and the general-purpose high-speed FIFO interface over USB.
For small runs, each populated board costs about £30.