With the wood chips, what is the likelihood that it's going to tie up a bunch of nitrogen and I'm going to have a hard time growing this year?

Slim to none ... any sequestering would have happened during decomposition and now it's over.

I think that the worst is over. If it was mine, I would add some nitrogen anyway to be sure.

Dont stop adding compost. The things you plant will tell you if they lack nitrogen, but I think you will be ok. During the growing season a few applications of fish emulsion will help nitrogen levels along with regular applications of compost. Observe & adjust as needed.

I know it's necessary to fertilize from time to time, but I always hesitate a little bit because of something I saw my dad do a few years ago. My parents have had a garden for at least the past 30 years and every year they buy plants from a greenhouse and plant the same things in the same place and to my knowledge have never amended the soil.

A couple of years ago, my dad pulled a good one and added manure to the garden and (since he doesn't really put much thought into long term garden care) thought "well if I'm supposed to add this much, then TWICE that should be better.

End result: Hugely bushy plants, two tomatoes harvested all year, and everything else was real low yield too. It's been two or three years and it still hasn't settled, although it is getting better.

Anyway, I know a soil test will tell me what i need to do, but I don't want to overcompensate for any real (or imagined) nitrogen loss and have that happen!

Your situation & the example involving your father are worlds apart.
The soil test will tell you all you need to know.

Few soil test labs today test for Nitrogen because we have found that N availability is temperature dependent. Most every soil test comes with recommendations to correct any deficiencies and most every one I have seen in the last 35 years suggested adding 2 pounds of N per 1,000 square feet, a fairly miniscule amount.
Most every garden I have looked at in the last 35 years that had problems had problems related to too much N although a few did show signs of N deficiency.
I would be less concerned about Nitrogen than I would be about soil pH and the balance of P, K, Ca, and Mg, drainage, tilth, Soil Food Web activity, soil smell, and the amount of organic matter.

Soil N tests are nearly useless. Soil N can fluctuate over the matter of weeks depending on influences. It's almost impossible to tell how much is plant available and how much is locked up...as well as how much will be soon volatilized, used by microbes in the soil to break down soil components making them further plant unavailable, and other uses of soil N influencing the pool of N.

The only thing general soil N tests are good at pointing out are extreme deficiencies or extreme abundance.

and extreme abundance was what your father had that year. I add nitrogen as needed, and it is easy to judge. Plants are dark green? none is needed. Plants are pale? give them some. Keep in mind that some plants (cabbage) almost always need some. Some other plants (carrots, beans) almost never need it. by using your eyes you can add it only where it is needed.

Pale green plant leaves might indicate a Nitrogen deficiency but also could indicate an Iron deficiency or a number of other nutrient problems or that could indicate a plant disease or possibly an insect problem. Then again there are a number of plants that simply grow pale green leaves.

Yes, but with the chips broken down, there is an abundance of everything other than nitrogen. One ton of chips is 2 lbs of P, and 6 lbs of K, and every other micronutrient that you can think of.