Warning: Long and detailed response
I have four kids ages 8-16 and they all started rafting and kayaking very young and over the years I've experimented with different ways to keep the warm on a budget, and it's definitely possible. Here are some ideas:
Buy used and resell when your kids outgrows the gear. Keep your open on ebay, Mountainbuzz, Boatertalk, Craigslist, year end close-outs on rental gear (shops, outfitters) and more for used gear for youth and small adults - farmer johns, neoprene (hydroskin) tops / bottoms, splash tops, drytops, booties, gloves, etc. I have bought 90% of the gear my kid's gear used, or on closeout for big discounts.
With all three options below, make sure to add neoprene head gaskets (or swim cap at minimum), neoprene booties and/or socks, and even gloves.
Good Option for Warmth:
Paddle Jacket / Neoprene Top / Farmer John wetsuit combo:
My kids spent many years in this combo. They hated drytops when they were younger, the tight latex neck can be very uncomfortable. My 11 year old still hates latex neck gaskets. This setup will keep your kid reasonably warm for several hours or longer, especially if they are actively moving (vs. sitting in a raft). My kids spent a lot of time swimming and cliff jumping in this setup both early and late season when the water and airtemps were cold. I always liked the padded knees of the wetsuits as the local river is shallow/rocky.
Better Option for Warmth:
Drytop / Neoprene or Fleece Top / Farmer John wetsuit combo:
This was the next step up for my kids, once they really started kayaking they learned to like the full drytop as it kept them dry when rolling. When my older two were comfortable rolling they'd usually opt for 1mm neoprene pants instead of the Farmer John wetsuit as the pants were easier on/off and more comfy. I would skip paddle pants unless they're sitting on a raft all day.
Best Option for Warmth:
Drysuit: difficult but not impossible on a budget.
You can usually get most of your money back on kids gear as long as you don't overpay to begin with, though it takes longer to sell than adult gear. There are some deals to be found if you're patient and have time. I have collected three for my kids. Two used ones I paid under $175 for each, including replacing several gaskets myself on each. I bought one new, a Kokatat Meridian (new old stock - T3 material) on closeout for $300 (less than 1/2 price) in adult size Small. Fits my 13 year old just right. Even on the cheap it was hard to justify buying the drysuits based on the dozen or so times they're used a year. But nice to have when needed.
Note: As I have upgrade my kids gear over the years I've held onto their paddle tops and dry tops. They gravitate towards those when the weather/water is warmer and/or depending on whether they are rafting or kayaking, etc.
Feel free to PM me if you have any specific questions.