@MikeG's answer is excellent. This is a supplementary answer only.
Part of the problem finding the coordinates of the "center" is defining the center. One example might be the center of mass, but finding all the mass is quite a challenge as we can't even see all of the galaxy's mass due to dust clouds, as discussed in How was the galactic plane established?, links therein and @RobJeffries excellent answer, and of course dark matter being invisible and all.
But even though you can't see and identify all of the mass, you can try to have a look at the way that all of the stars that you can see seem to be moving via their radial Doppler shift, and at least try to estimate it.
However, for the purposes of your question, I'm not sure if that's the only definition of "center" that could work for you.
If instead you'd like to consider something more catchy and visual, how about the supermassive black hole that's almost for sure at the "center" of our galaxy, affectionately known as Sagittarius A*? That's just Sgr A* to her friends (see also Why is Sagittarius A* called so?). According to that article, the distance from our Sun to Sgr A* is 7,860 ±140 ±40 pc, which is a remarkable thing in itself because it has error bars on the error bars!
That number is linked to the ArXiv preprint An Improved Distance and Mass Estimate for Sgr A* from a Multistar Orbit Analysis, which is also an analysis of how some very special stars move.
Here's a GIF, found in this excellent answer to What is the evidence for a supermassive black hole at the center of Milky Way?
The reason that I call this a supplementary answer is that you end up with very nearly the same answer as @MikeG. Here I've used the same procedure:
X Y Z
parsecs: -430±8, -6861±122, -3812±68
lightyears: -1401±30, -22,379±399, -12433±221
AU: -8.86(0.16)E+07, -1.415(0.025)E+09, -7.86(0.14)E+08
km: -1.326(0.024)E+16, -2.117(0.038)E+17, -1.176(0.021)E+17