Does Summer Savory Reseed Itself?

I would like to know if summer savory will seed itself? I planted in a planter box last year. Now I have many "sprouts." I am not sure as to what they are yet.

Yes, summer savory can reseed itself. It is an annual, so it normally reproduces itself when overwintering seeds sprout in spring as the soil warms.
Of course, you could have weeds coming up instead. Once the true leaves form – these are the leaves formed after the initial pair – you should be able to recognize whether or not the seedlings are likely to be summer savory or not. If the leaves are narrow and positioned in pairs opposite to each other along the stem then you vary probably have summer savory coming back.