The Fed has already been buying ETFs, which is about a hair’s breadth away from buying stocks. I don’t doubt that they’ll buy stocks to prop up investors if the current stimulus proves ineffective.
Barely. They are only buying corporate bonds and the sum total of "Includes non-marketable U.S. Treasury securities, supranationals, corporate bonds, asset-backed securities, and commercial paper at face value" is 85.274 billion or a max of 1.2% of their asset sheet [0].
The possibly essential caveat being that the ETF's being purchased are (as far as publicly known) all bond ETF's.