A single solvent? Probably not. Different elements have dramatically differing solubilities in most solvents, and so you would have to tune the solvent to the element at least to some extent.
However, for the specific task of identifying a solvent that would (a) burn and (b) dissolve chemicals suitable for giving either a blue/green color or a pink/violet color, then yes, I think it should be feasible.
Aqueous ethanol solutions at a concentration above about $20\%$ w/w are readily flammable at room temperature, and the aqueous base should provide a suitable base solvent for many salts. Vodka might be one good readily-available source of an ethanol-water solvent for this purpose.
To achieve the pink/purple color, potassium chloride should be a good option, as it's freely soluble in water and the potassium gives a purple color in the ethanolic flame.
Getting the blue color might be a bit trickier, as the most common element giving a blue flame is copper, and most copper salts aren't especially soluble in water at neutral pH, or in ethanol. What might work is to acidify the ethanol solution with a high-concentration vinegar stock (I've seen products available at up to $45\%$ acetic acid), and then add the copper salt. Copper sulfate pentahydrate is probably a good option there, as it's commercially available as, e.g., an algaecide. You'd have to experiment with the recipe to find something that burns readily with the desired color, but that doesn't smell too strongly of vinegar.