Or cut the roots off outside the beds. Three hour, and 2 blister job with a pick axe.

Or start new beds. Do you have a sunny area devoid of trees?

Unfortunately we live in a forested area. The beds are in the sunniest area. This problem had been creeping up for many years as the garden is about 25 years old.

Unfortunately, the Spruce trees are doing exactly what they are supposed to do: take over the shallow rooted, easily destroyed meadow and turn it into stable sustainable forest. They are letting your vegetables know that their services as annuals are no longer required.

The only solution is to take bunnydefluff's suggestion and make like a pioneer and clear a piece of forest for your vegetable garden.

Many of us have similar problems even in suburban yards where we planted trees years ago that are now invading the garden 10 or 20 or 40 feet away. If you dig down along the edge of the bed you can cut them, but if they have never been cut they may be as big as your arm so be prepared. Once you get the big ones, if you come back every year or two they will be small and fibrous and much easier to maintain.

It will not solve any problem you may have with deeper roots sucking moisture from below. If it seems like you can't get enough water on even after removing roots you can get to, it may be that. But it depends on the species, soil, climate etc. so you may not have such a problem with spruces in your locale.

A pick axe is good for cutting roots when you get to them, or the other good gardening tool for digging is a pick mattock. It has a heavy chopping blade good for loosening tight soil and rocks, and a pick for really dense soil or prying out bigger rocks. I got a nice long handled one from easydigging.com (no affiliation) that has been an indispensable tool.