In May 2017 I proposed an MAA pdf ebook series organized by subject. My proposal included a sample covering the years 2000-2016, a subsample of which can be found here.
The proposal was received well, but it ultimately got rejected because nobody at the MAA knows whether it has the authority to reprint such a collection without obtaining new permissions from all the thousands of published proposers and solvers. It probably does not help that Taylor & Francis now owns the journals too.
After receiving the rejection I sent each lead editor of the three problem columns a full version covering 2000-2016 (including the Putnam), organized by subject. One of them replied: "Thank you very much for this incredible work! This is very useful. Is this open to everybody out there?"
Sadly, no.
Earlier attempts include an ambitious database project spearheaded by Ivars Peterson during his tenure as Director of Publications (2009-2014). It eventually met with powerful resistance and died.
The MathPro Press collection (see Jonas's answer) was once available in a free searchable database. Snapshots of the 1999-2005 and 2005-2010 versions are here and here, respectively.
I am currently putting together a free pdf of Index to Mathematical Problems 1975-1979 for my ResearchGate page. Will put a link here when it's finished. Stanley Rabinowitz is interested in doing the same for the 1980-1984 index.
============================ added 2018 Aug 16 ============================
Update:
The 1975-1979 index is now freely available as a pdf on my ResearchGate page and on my website, here:
http://www.mathematrucker.com/problems_1975-1979.pdf