Warning: Bromates are suspected carcinogens. Limit exposure to them. They are also strong oxidizers. Keep away from combustible materials. Bromine is smelly and toxic in large quantities. Perform this reaction outdoors.
You will need: Source of potassium (No Salt salt substitute, Spectracide stump remover, etc.), bromide reserve for spas (sodium bromide), a carbon rod, a wire of any sort, a five volt power supply, and some connecting wires.
Boil some water and dissolve quite a bit of sodium bromide in it, though not enough to make a supersaturated solution when cooled. Anchor the carbon rod and the bit of wire to the sides of the chosen electrolysis container and pour the hot sodium bromide solution in. Attach the positive electrode of the power supply to the carbon electrode and the ground to the wire. Begin electrolysis.
A brown solution of elemental bromine will begin forming at the carbon electrode (anode), while a colorless solution of sodium hydroxide will begin forming at the wire (cathode) (net reaction 2 NaBr + 2 H2O --> 2 NaOH + H2 + Br2). The hydrogen escapes as a gas from the cathode. When sloshed around, these two solutions will react to form sodium hypobromite and sodium bromide (2 NaOH + Br2 --> NaBrO + NaBr + H2O). Because of the heat, the sodium hypobromite will disproportionate into sodium bromide and sodium bromate (3 NaBrO --> NaBrO3 + 2 NaBr). All of the sodium bromide will reenter the reaction at the beginning.
When bubbles begin forming at the anode at a significant rate, this means that much of the sodium bromide is exhausted. Stop the electrolysis and filter the solution to remove the carbon particles. What remains is a light yellow solution of sodium bromate, bromide, and hypobromite. It smells just like household bleach, just with chlorine instead of bromine. Heat it to disproportionate the majority of the remaining sodium hypobromite, then let it cool. When crystals begin forming (it may take several days of evaporation), add some potassium salt solution and stir to dissolve the crystals. They should dissolve but new crystals should take their place over time. These are crystals of not-so-soluble potassium bromate. Wait until no more crystals form and then remove them. They should be quite pure and white. The remaining solution contains some bromate in solution as well, but it is quite impure. When I tried this, the yield was not spectacular, but the resulting crystals were good. You can use them as an oxidizer or a source of reactions involving bromine oxyanions.