Gillian,
Try to get another sitter first. If that's not possible, try whitepages.com and look up the bride or groom's name and give them a call to say your sitter fell through and you'll send their gift. If they don't mind if your child goes, they'll tell you then. But they are paying for your dinner whether you're there or not so a call to them would be courteous. If they aren't listed, think of a mutual acquaintance or friend who might have their number and go from there.
That said, I COMPLETELY DISAGREE with people who said young children are not welcome at weddings. I read all the replies and I feel bad the one lady's day was ruined by her thoughtless brother and SIL but it sounds like the child was welcome--the parents are the ones who behaved poorly. I think the others who said "no children at all" are being polite . . . some folks definitely do not want kids at their wedding but some people truly don't mind having children attend. The flower girl and ring bearer aren't hidden in closets until it's time for their duties and then put back, are they?
If you know your child would behave well during the ceremony and reception (and naps count as good behavior!) lol let that be your final factor in whether or not you look for a sitter, see if she can attend or decide that one of you will attend alone.
I think you're very kind to consider their feelings in this. Just know that some people would be more disappointed if you didn't go than if you ended up bringing a well-behaved child.
D.