Maybe I don't quite understand the question, or it has particular nuances in America that I am not aware of.
From my standpoint here in the UK, straw is a natural product and I would, have and will dig it in, if a rototiller will not do it, my spade and a bit of graft will. Similar to turning green manure in, I would just dig spade deep and turn it over so the straw is now approx. 9 inches down, the soil will now deal with it and as I am on a clay loam soil the straw helps to separate the clay molecules and thus aid drainage.
I'm in a rural part of the UK and we get and use a lot of cow manure, a huge proportion of that is straw and it is spread on our beds over winter, dug in, in spring with no issues. If it is dry and matted, either wait for rain or wet it...job done.
I would stay away from covering productive veg beds over winter with wood chips because it uses massive amounts of nitrogen as part of it's decomposition process, robbing your soil of this valuable and essential nutrient that your beds will need early spring for all that initial healthy green growth of your plants, unless of course you intend to supplement with fertilisers.
Steve...
