They are both correct. 'Loud' can be both an adjective and an adverb. One of the great myths about English grammar is that adverbs - words describing verbs - always have to end in 'ly'. This is completely untrue. In fact, we have many adverbs, known sometimes as 'flat adverbs', which don't end in 'ly'. Other examples of flat adverbs are 'tight' ( as in 'Hold on tight'), 'straight' ( as in 'Stand up straight') and 'right' (as in 'Do it right'). In some cases, there is no 'ly' form of the adverb. For example, 'fast' and 'well'. In other cases, there is both an 'ly' form and a flat form of the adverb, but they have different meanings: for example 'He works hard' and 'He hardly works' mean very different things! And in a handful of cases, there is both an 'ly' form and a flat form of the adverb, and they have the same meaning. This is the situation with 'loud' and 'loudly': they are both equally correct.

"I clap loudly", as it follows the structure "subject + verb + adverb"

I believe it's "loudly" because it refers to a verb, not to a noun.

2. I clap loudly. - I would say it this way first, if I'm being formal. If I was speaking casually, I would say, "I clap loud".

2. I clap loudly