Learn Spanish (Spain): Ellas aceptarán el trato a condición de que abonemos el anticipo. - They will accept the deal on the condition that we pay the advance.

The phrase a condición de que (on the condition that) sets a strict requirement for the main action to happen. Because it refers to an action that hasn't happened yet (and might not happen at all), this type of conditional linker always triggers the subjunctive.

Could I drop the que and just say a condición de abonar?

Not in this exact sentence, because there are two different subjects: they (ellas) accept the deal, but we pay the advance. You can only drop the que and use the infinitive if the subject remains the exact same for both actions.

Is there a difference between using abonar and just using pagar?

Abonar is a slightly more formal synonym for pagar (to pay). It is very frequently used in business, finance, and contracts to refer to paying deposits, installments, or advances (el anticipo). Paguemos would also be grammatically correct here, but abonemos sounds more professional in the context of a deal (un trato).

Could I swap a condición de que with other phrases from this lesson, like con tal de que?

Yes, they are highly interchangeable here! You could easily say con tal de que abonemos, siempre que abonemos, or siempre y cuando abonemos. They all express a strict condition (translating to "provided that" or "as long as") and all require the subjunctive.