What sort of fallacy is this?

From here:

So, the votes of 75 percent of Oklahoma’s citizens are invalid, because the “emerging awareness” doctrine declares that 5,000 years of human tradition constitute “an arbitrary, irrational exclusion.”

 

Is this an appeal to tradition? I mean he tries for special pleading against gay marraige because "it's tradition" despite the fact that the legal rights were what was at stake, not any religious definition. 

I would go with the fallacy of equivocation. Defining marriage religiously and then switching to the legal definition. Which is simply a matter of a contractual agreement between two adults and has nothing to do with religion.

 My argument has been that......

If you refuse to issue a marriage license based on sex, you've violated federal law that prohibits states from discriminating against anyone based on sex. Then not issuing a marriage license based on sex is illegal,

Therefore it can't matter if two men or two women want a marriage license. So in all actuality, "same-sex" marriage has been legal since antidiscrimination laws have existed.

At first glance, I smell obfuscation. What the hell are they talking about???

First of all, I'm a Christian and I think that marriage should be between a man and a woman. With that said, I think you're right, the law amendment in Oklahoma was declared unconstitutional due to the 'emerging awareness' doctrine. This would mean that the reason why the amendment was adopted was because of tradition. That is the grounds of the doctrine as it states

"In all events we think that our laws and traditions in the past half century are of most relevance here...."

This is why there is an 'emerging awareness' as in, awareness was needed to issues pertaining to adults "...in deciding how to conduct their private lives in matters pertaining to sex," also found in the ruling. 

Therefore, the Oklahoma amendment was declared as unconstitutional because the amendment was 'based on “moral disapproval”' and stated that 'marriage in this state shall consist only of the union of one man and one woman,...'

I agree with your assessment. This is an appeal to tradition because the preferences of 75% of Oklahoman's citizens was reflected on the voting record and this record was used to adopt the amendment by referendum. The 'emerging awareness' doctrine disagrees on the correctness of such ruling as the doctrine 'declares that 5,000 years of human tradition constitute “an arbitrary, irrational exclusion.”'

I can't just leave this comment without mentioning what the Bible says. This would not be an appeal to tradition but a cry to awareness. 

Or do you not know that wrongdoers will not inherit the kingdom of God? Do not be deceived: Neither the sexually immoral nor idolaters nor adulterers nor men who have sex with men nor thieves nor the greedy nor drunkards nor slanderers nor swindlers will inherit the kingdom of God.
1 Corinthians 6:9-10

The "5000 years" part of the argument is appeal to tradition .

The phrase "the votes of 75 percent of oklahoma's citizens are invalid" is a bit inaccurate. What is it actually is that "the votes of 75 percent of oklahoma's citizens are irrelevant when they collide with constitutional rights/equal rights)". It's like saying that 75% voted in favor of discriminations, therefore if we don't have discriminations their votes are invalid." They clearly don't understand equal rights or democracy, but that's not a fallacy.
I don't know if they purposely used that inaccurate word or not. But if they did, it could be argument by emotive language (in the sense that people get angry when they feel invalidated).

Before answering the question presented, there are a few preliminary questions to resolve. Are all those 75% invalid votes living and voting? Or are they dead and voting? Or should we grant reparations votes to the dead who may not have voted in the 2020 elections and give them three votes each? It probably doesn’t matter because there are enough corpse voters to make whatever should be a clear choice election in doubt. Oh, by corpse voter, I don’t mean a dead person voting. I have friends who are proudly corpse voters. By which, I mean a person voting for a corpse if the toe tag reads: Democrat.