It is correct to use “Looking forward to meeting you soon” because the -ing verb (“meeting”) there is a gerund and functions as a noun. “Looking forward to meeting you ...” can be replaced with “Looking forward to our meeting ...” where the word “meeting” is a noun.
Another example of this usage can be seen in this sentence: “I’m not used to sleeping late” which can be replaced with “I’m not used to late nights.” In this sort of structure, “to” is a preposition and not an infinitive marker, as in “I would like to meet you soon” OR “I do not like to sleep late.”
An “-ing” is placed after the verb to make the present participle of the verb, which is used in the continuous/progressive forms of the verb, e.g. “I am eating”, “I was eating”, “I shall be eating”, etc. The present participle can also function as an adjective, as in “Managing Director”, “running commentary”, “a boring film”, “an exciting match”, etc. The only instances when we can use the “to ...ing” structure is when the “-ing” form is a gerund.