Is there a "Reina-Valera" only movement? I'm aware of the King James (KJV) only movement, but is there also a similar movement for the Spanish Reina-Valera (RVR) bible? My father, who is Puerto Rican, told me that RVR-onlyers were a thing.
Yes there is.  And like the KJV-only movement, it is not monolithic.

For English speakers, it's important to recognize that the Reina Valera (RVR) is a group of Bible translations that continues to dominate the Bible translation market in Spanish-speaking countries. They primarily rely on the Textus Receptus, like the KJV, not modern critical Greek texts, like popular English translations (NIV, NLT, ESV, etc.).  The most common edition is the 1960, but the two other 20th century editions, 1909 and 1995, are also popular.

With that in mind, we can break down "RVR only" tendencies as follows:


Those who prefer some edition of the RVR because they are most familiar with it or it is most common in their area
Those who generally prefer the RVR because they consider the Textus Receptus to be the best basis for a Bible translation
Those who consider one particular version of the RVR to be the "correct" or even "inspired" translation of the Bible into Spanish


The KJV-only movement can be broken down similarly, but the distinctive in the RVR-only movement is that not all those in group 3 agree on what is the "right" version of the RVR.  There are at least three versions of the RVR that are claimed to be the only "correct" version:


1602 Purified.  Robert Breaker III examines the various RVR options and concludes that a recent revision of the original 1602 RVR is the "purest" Bible translation. (source)
RVG (Reina Valera Gómez 2010).  Emanuel Rodríguez wrote a book entitled, God's Bible in Spanish: How God preserved his words in Spanish through the RVG.
1865. Jeff McArdle argues that the Valera1865 is the best representative of the original 1602 text, saying that "every attempt should be made to defend every word found in the Valera 1865." (source)  


In this sense, the RVR-only movement is even more divided than the KJV-only movement.  These three camps naturally criticize each others' versions; for example, Emanuel Rodríguez (the proponent of the RVG) issued a lengthy rebuttal against Jeff McArdle's criticisms.

It's worth noting that the RVR-only movement seems to originate in, or at least get much of its voice from, the KJV-only movement.  Robert Breaker and Jeff McArdle are both fundamentalist baptists and KJV-only proponents, and a prominent KJV-only advocate, D. A. Waite, endorsed the RVG and hosts Emanuel Rodríguez's defense of it on his website.