When our three-year old son was hospitalized with a serious medical condition, we sought the anointing of the sick for him. We were told that it was against Church policy to anoint children in these circumstances. He recovered, but should something like this ever happen again, we would like to know what the rules are.
While I wouldn’t have used the term "Church policy" to explain the withholding of the sacrament of the anointing of the sick from your son, the answer you got seems basically accurate.
Canon 1004 states that anointing may be administered to those who have "reached the use of reason." Church law presumes the use of reason to set in at the age of seven (canon 97), although it is possible that, in a specific case, a given child might attain the use of reason prior to turning seven. Nevertheless, to say that a child has the use of reason at age three would be, I think, virtually impossible to prove.
Canon 1004 also states that one must "begin to be in danger of death" in order to receive the sacrament licitly. It is quite possible that your son’s condition, however dangerous, did not encompass the danger of death, and for that reason the sacrament would not have been celebrated. The canon does not require that one be at death’s door before seeking anointing, but death (as opposed to any other chronic but survivable condition) must be a genuine possibility before administering the sacrament.
Your instincts in seeking the Church’s spiritual and sacramental support on behalf of a stricken child were correct. In seeing to your son’s baptism, you already afforded your toddler the grace of that saving sacrament, and, in turning to the Lord in prayer during this crisis, you already invoked additional graces on his behalf and yours. Moreover, had your child been somewhat older, I am sure you would have inquired for him about confession (canon 989) and the Eucharist (canon 913) as well.
Lastly, you didn’t ask about this, but I would mention one other possibility. If your child had been in danger of death, and notwithstanding his ineligibility for the sacrament of anointing because of his age, he very likely was eligible to receive the sacrament of confirmation (canons 889 and 891). I know this comes as a surprise to many Catholic parents (and to not a few clergy!), but the 1983 Code of Canon Law is quite clear that confirmation is proper under the circumstances you describe.