The integrated IGP of the 5600GT processor uses system memory to assign for VRAM. Therefore the amount it can allocate to use depends on the total amount of system memory present. It also depends on the motherboard support.
First make sure you are on the latest non-beta BIOS release. Then enter the BIOS under Advanced Mode (not Easy Mode) and go to Settings, IO Ports. Select UMA Mode and change from Auto to UMA Specified, and set the UMA Frame Buffer Size to the desired size.
I would urge you to benchmark test between 2GB and 4GB of VRAM however, as you will likely notice very little if any performance improvement in gaming. The IGP is simply not powerful enough to need or benefit from more than 2GB of VRAM. In fact, there is only a marginal improvement going from the default 512MB to 2GB. (<10% or around 1FPS on average in testing)
As for choosing a future GPU, keep in mind that the 5600GT processor only supports PCIe gen 3.0 and only 8x lanes for the graphics card.
Ryzen R7 5700X | B550 Gaming X | 2x16GB G.Skill 3600 | Radeon RX 7900XT
Ryzen R7 5700G | B550 Gaming X | 2x8GB G.Skill 4000 | Radeon Vega 8 IGP
Ryzen R5 5600 | B550 Gaming Edge | 4x8GB G.Skill 3600 | Radeon RX 6800XT