Can you use a mana from a land with an activated ability to pay for a spell if that ability requires tapping?

Absolutely. Tapping a land to produce mana is a mana ability (CR 605.1a), and mana abilities can be activated any time you have priority and need mana — including during the process of casting a spell (CR 601.2h). The tap symbol ({T}) in the cost simply means you must tap the permanent as part of paying the activation cost, which produces the mana.

When you begin casting a spell, after announcing it and making all required choices, you get a chance to activate mana abilities to pay the total cost (CR 601.2h). This is the window where you tap your lands. The mana produced goes into your mana pool and is then spent to pay the spell's cost.

Importantly, mana abilities don't use the stack (CR 605.3b), so they resolve immediately. This means no one can respond to you tapping a Forest to get green mana before you pay for your spell. The mana is simply added to your pool and used right away.

Example: You want to cast Llanowar Elves (cost {G}). During the casting process, you tap a Forest using its activated mana ability ({T}: Add {G}). The Forest is now tapped, one {G} enters your mana pool, and you spend it to pay for the Elves. The spell is fully paid and placed on the stack.

One condition: the land must be untapped when you want to activate the ability, since the {T} cost requires tapping an untapped permanent (CR 602.2a). A land that is already tapped cannot be tapped again to produce mana.