Let's say that I am in melee with an enemy. On my turn, for one reason or another, I take the Disengage action but do not move out of melee range.

In the same round, the enemy starts their turn by moving out of my reach.

The PHB says [emphasis mine]:

If you take the Disengage action, your movement doesn’t provoke opportunity attacks for the rest of the turn.

The phrase "the turn" is ambiguous as to whether it is only my turn.

According to RAW, does the Disengage action negate my ability to take an Opportunity Attack? Should I get an Opportunity Attack in this situation?

Yes, disengage does not prevent you from making opportunity attacks, only provoking them.

If you take the Disengage action, your movement doesn’t provoke Opportunity Attacks for the rest of the turn.

Only your movement doesn't provoke Opportunity Attacks. Everybody else's movement provokes as normal.

Since the enemy's movement still provokes an opportunity attack, they provoke one from you which you can take just like normal.

What is a turn?

Even though it doesn't matter for your case here, a turn is the time when your character can move and/or take actions. When they are done with doing those things their turn is done. A round is the series of every character's turn from the first person to the last. After the last person's turn ends, the round ends and then begins anew.

In this case, you take the Disengage action on your turn and then you end it by an enemy. At this point, Disengage no longer has any effect since your turn is done. Your enemy's turn starts and they move away from you, provoking an Opportunity Attack from you according to the normal rules.

Yes, you do.

First, a round is an interval of time. A turn is the procedure for handling a single character's actions during a round. You take the Disengage action as part of your turn, then your turn ends. The enemy acts on its turn.

Second, Disengage says "your movement" doesn't provoke opportunity attacks. It says nothing about anyone else's movement.

Your Disengage action won't stop you from taking opportunity attacks

Firstly, this is because when you Disengage (PHB p. 195, bold added):

your Movement doesn’t provoke Opportunity Attacks for the rest of the turn.

It doesn't stop anyone else's movement from provoking those attacks. So even if someone moved during your turn after you had Disengaged (say, if they had a held action to run away, and took it during your turn), you would still get an opportunity attack on them.

Secondly, the Disengage action's effects only last until the end of your turn, not the end of the round. So even if it had worked the way you thought it did, its effects would not be in play on a different creature's turn.

There is no actual link between your PC's opportunity attacks and another character's, it's just that very often you are both within each other's reach. As the PHB (195) states:

You can make an opportunity attack when a hostile creature that you can see moves out of your reach ... You can avoid provoking an opportunity attack by taking the Disengage action.

So long as you have not already used your reaction, and the other character is not disengaging, you may choose to attack.