Pretty much any pattern can be re-sized, but it may take some serious math. A good place to start is with the child’s measurements. This pattern has sizes for 3, 6, 12, and 18 mos, and the chest measurements are 16, 17, 18, and 19" respectively. Looks like they’re adding 4 CO stitches for every 1" the pattern jumps up in size/chest measurement. According to standard sizes, a 3 year old has about a 22" chest. That would mean you’d need to add 12 stitches to the 18-month-size CO. (22" - 19" = 3" x 4 stitches = 12)
There are several other things that will come into play adjusting the pattern–length, where to start decreases and how many to do, etc. It can get complicated, but the good thing about knitting is that if it’s not working, you can always frog back and try again! Good luck–and congrats on getting a “commission”!