Taylor was the pantomime villain at Upton Park in each club's final game before Christmas, showing needless red cards to both Hammers striker Carlton Cole and Toffees midfielder Darron Gibson.
Both were sent off for dangerous play after catching opponents while raising their legs in respective attempts to control and win the ball.
Allardyce accused Taylor of being trigger-happy, saying: "It is how quickly the referee got the red card out, he couldn't wait, he didn't deliberate over the decision or anything.
"There is nothing much we can do about the result but we can appeal the decision and hopefully we can get Carlton down to a yellow card and get him off."
Moyes added: "I will appeal it. "I'll have a word with Sam as well, because I don't think you could put the word frivolous next to that appeal.
"Because I think you would say there was genuine reason why maybe you could appeal that.
"Now, you don't want to go and appeal and find out that maybe you get an extra game on it. That's what you don't want to do."
Allardyce was planning more than an appeal, however.
"I spoke to the ref but that is between me and him," he said.
"We will go through the channels of 'reporting' that the system has in place and I will take that up and pursue it quite vigorously.
"If he is not getting the right feedback or coaching and is told it was okay and 'Carry on' then we have got a big problem."