It is totally possibly to use VHDL and Verilog with all Alchitry boards 
For the boards with Xilinx FPGA (Au and Pt), you can use Vivado to write code and program the FPGA, but because Vivado restrict the compatible flash chips for unknown reason, you will need to use the Alchitry Loader to program the flash.
It can happen that the board is not detected by Vivado but there is a solution here.
For the Cu boards, you can use either the open-source toolchain IceStorm or Lattice’s IceCube.
The only caveat (except the flash) is that in these IDEs, the boards are not defined so you will need to map the pin names from the connectors to the FPGA package pins using the pinout files : Alchitry-Labs-V2/src/main/kotlin/com/alchitry/labs2/hardware/pinout at master · alchitry/Alchitry-Labs-V2 · GitHub