Relay-based scanners are common because they are the most versatile due to the high insulation resistance in off state. Obviously switching/settling delay and relay wear is an issue. Some multimeters (e.g. Keithley 199/2000/2001, some Prema meters, HP 3457A) have an optional scanner/multiplexer plug-in. These are often limited to 8-10 channels. There are also stand-alone multiplexers for if you want more channels/flexibility.
The HP/Agilent/Keysight 34970 was already mentioned. Keithley has the older 700 and the newer 7000 and 2700 series. HP had the 3488a. Make sure you get a matching scanner card.
There are a couple of scanner cards using solid state relays. The
Keithley 7710 is one such card for the 2700. There is currently one
on ebay with 2700 for $540 or best offer, which is not a terrible deal in my opinion (no association to the seller, can not vouch for the seller or this unit). Note that the 2700, unlike a 7001, includes a multimeter.
The
Keithley 7015 (-C and -S only differ by the connector type) is another solid state scanner card, but for the 7001 mainframe. Did not see any affordable ones on eBay right now. Not sure what Agilent/Keysight solid state scanner modules might be affordable.