First question, do you have a back-up glove?

I've re-laced only a few fielder's gloves, years ago. It is pretty simple, but it is time consuming the first time or two. Like Coach advises, I used one of those big needles. I also used a pair of hemostats for some really tight places where the needle and leather wouldn't both fit.
Unless there was a broken lace, I would start where the knot is and pull the old out and follow it with the new,one segment at a time. It can be tricky doing it this way, especially if the hidden laces are tight and run adjacent to a different lace. This is another spot where I used the hemostats.
Patience will get you through the process.
I used to have a Tandy leather store nearby and bought my lacing there. I sometimes found that the lacing packed with the needle I bought at a sports store, is inadequate for all applications. For a web,I would use the thicker stuff.