Translation:The buyer is paying for the horse right away.
Correct in the literal sense - but nonsense in meaning! I'm just curious to know how you would say "The customer pays the man for the horse" in German? Would the horse be dative then? If so, why is it not dative in this sentence too? It is implied as the indirect object, and with the "for", it is dative in the English.
To your question: you put "for" in your English sentence. The same happens in German.
"Der Käufer bezahlt fuer das Pferd sofort." that's fine.
You can say in German for "to pay for something which he just bought":
Er bezahlt fuer... Er bezahlt das... Er zahlt fuer... Er zahlt das...
If someone did a service for someone, you can say:
Er bezahlt dich dafuer, Er zahlt dir deinen Obolus/Betrag/Lohn/Gehalt/Gage, Er bezahlt den Arbeiter
Er bezahlt das Pferd.
Hope that is enough explanation. If there would be some context given, this confusion wouldn't have started.
Some addition: In everyday German you may hear the question:
Hey, hat er schon das Pferd bezahlt?
Now you may think you can answer: "Ja, er hat das Pferd bezahlt" or for our example: "Ja, der Käufer bezahlte das Pferd sofort." or similar.
With objects it would be fine, with persons not, see above. A horse might be an work horse, that makes it ambiguous, as someone else posted here.
"Der Käufer bezahlt das Pferd sofort." = ambiguous
"Der Käufer bezahlt das Paket sofort." = possible
"Der Käufer bezahlt fuer das Pferd sofort." = good
I don't think it would really a synonym anyway; to me, shopping is an activity which basically involves looking for things in a shop/shops (either specific items or generally) with the intent to purchase them, but does not include the actual purchase. This is sort of similar to the distinction between seller (one who sells) and salesperson (one who attempts to persuade people to purchase a product or item, usually on another's behalf). A salesperson may also be a seller, but the two are not synonymous.
Related questions have been asked before; for me, I can only understand it by thinking that there are two English meanings for "bezahlen" - to pay, and to pay for. "The buyer pays the horse" may be a word-for-word version of the German, but it does not make sense in English - unless you actually employ the horse as analogkid suggests above! It has to be "The buyer (or customer) pays for the horse". Hope that helps.
Is there a reason that 'The buyer purchases the horse right away' is wrong, but 'The buyer is paying for the horse right away' is right? Is there a different word or grammar for 'purchases' vs 'is paying for'? I just want to make sure 'bezahlen' means the right thing in my head.
My guess is it's just like in English - "paying for" is merely the act of handing over the money while "purchasing" also includes the receipt of the item being paid for. "Kaufen" means to buy (which, unlike "pay for" is synonymous with "purchase"), while a quick look on Wiktionary shows that "anschaffen" may also work for purchase (although I haven't come across it before now so can't comment on its usage).