How do I convert volume to weight for soft cheeses like ricotta and mascarpone?

I don't have mascarpone to check, but the Internet says that there are on average 225 grams per cup. Mascarpone being mainly a mix of water (density 1 g/cm^3) and fat (density 0.9 g/cm^3), this sounds about right.

I don't think it is possible to give a reliable conversion for ricotta. First, many countries don't have normal ricotta, so people use all kinds of substitutions, frequently not being aware that they are not the same thing. So it is possible that whatever you can buy in your supermarket under the name "ricotta" is not the same thing as whatever the recipe author can buy. Second, real ricotta is made from whey, and the amount of whey left in the cheese varies, so you will have a different amount of water between brands, which will change the density.

With this difference, I guess you could just assume that ricotta has the density of 1, measure out 235 grams and call it good enough. This will usually be a different amount from what the recipe author uses - but then, so will be measuring out a cup by volume.

Most American recipes use volume instead of weight for measurement, which means there's often not an exact measurement in weight for the recipe. Your best bet would be to find out what kind of cheese it is calling for, find the density of that cheese, and then go from there. Also whether the cheese is shredded, still in a block of some sort, or if it's a wet and clumpy cheese like ricotta will play a significant role in density as well.

I would suggest getting a measuring cup, measure and weigh a cup of cheese several times, then average out your measurements so that you can just use weight next time you want to do the recipe.