According to The Atlantic, there have been 240 bills proposed in 2013 that were represented by an acronym, such as the ROBO COP bill to help curb automated robot calls and the BEER (Brewers Excise and Economic Relief) act. Those have since remained very common, as evidenced by the recently written SAFE and HEAR acts which are currently in the legislative process. A famous less recent example is the PATRIOT act passed in the wake of 9/11. When was the first time that a house bill in the United States was named as a meaningful (dictionary word) acronym? How long did it take for more legislators to start using them?
While there are a few early examples from the 1970s and early 80s, these seem to refer to existing acronyms. For example there was the "ACTION Domestic Programs Amendments 1973", but its not clear that "ACTION" is an acronym at all. Similarly there is the "Port Authority Trans-Hudson crisis prevention act 1981" (PATH) but this refers to a pre-existing acronym. It is difficult in these early examples to distinguish a catchy acronym coined by a representative from one that already existed.
By the late 80s, catchy acronym titles were sometimes used, for example, the "Comprehensive Alzheimer’s Assistance, Research, and Education Act of 1987" (CARE) The use of acronyms has risen since the late 80s, slowly from 87-96, and then faster until now about 6% of bills use some kind of acronym.
All information and stats are taken from https://noahveltman.com/acronyms/