One of my players is playing a yaun-ti pureblood ( essentially a 30 ft long snake person) and the party will be headed into a rather large port town next session. I was planning on giving them a disadvantage on things like persuasion and advantage on other stuff like intimidation
All that being said, yes it is perfectly appropriate to have NPCs react differently based on the race of the player, particularly for those "evil" races. That is part of the interesting story of a "monsterish" character. If you and the player have come to an agreement that the character looks *very* snakey, and that snake-folk are not well regarded, then that will definitely impact on how they handle social situations.
However, I would not implement that via blanket disadvantage on rolls - that feels like punishment. Better to adjust the DC of checks based on the NPCs personalities. Say there is an NPC and it is DC15 to convince them to do something, but there are different ways to do that. This NPC has high wisdom/insight so maybe trying Deception would be more like DC20. Now when the snake person enters maybe this NPC hates and fears it, so now a Persuasion attempt might also become DC20, but at the same time an Intimidation effort could be only DC10 coming from a snake due to that fear.
If your player's Yuan-Ti Pureblood is more of the described human-looking infiltrator, however, then no change to social interactions unless someone discovers their true nature. The game for the player then becomes avoiding exposure - and you better bet there is some pesky Ranger sniffing around for secret snake spies...
She's a pureblood but through magical means she chooses to have a snake tail instead of legs
Ok, so the character looks like a monster. NPC commoners should treat them like a monster. Forget disadvantage on persuasion, there should be running, screaming, throwing stuff, and mobilization of the town guard.
At least in the forgotten realms setting, monstrous races are not considered "people" by humanoid governments. There are rare exceptions where a village might have a truce or trade agreement with humanoid monsters in the area, but even then common people will be cautious, and it would only apply on a case-by-case, village-by-village basis.
How are "snake people" viewed by the general populace? If people think of them as "monsters" they will react accordingly and may well flee or attack before the character even has a chance to speak. If they are viewed as "people", even if that is scary people, the character may get that chance to speak, probably with a higher DC, or possibly with disadvantage.