The ADC's made for audio usually are rather bad at gain stability and thus not very useful for a meter.
There are some cheap sigma delta ADCs, that may be useful in a cheap meter: e.g. MCP3421 (18 Bit), LTC2440 (24 Bit), ...
However the resolution quoted for sigma delta ADCs is usually the limit where you get no missing codes. A reading on a LCD should be reasonable stable and linear. So you want something like less than about 0.2 digit DNL and noise. So you can't just convert LSB steps of these ADCs to display steps. So a 24 Bit ADC like the LTC2440 may be good for a 5 1/2 digit or at most 6 digit DMM. Still the ADC is just one part, and not the critical one any more.
The dual slope converters like ICL7106/7 found in old 3,5 digit DMMs have only about 12 Bit resolution, but something like 0.01 LSB DNL. So even if you have a normal 16 Bit ADC and suppress the lowest 4 Bits, leaving only 12 Bits, the DNL errors are likely larger. In these old days adjustments were done analog and thus there was a direct relation from ADC steps to display. Modern meters tend to use digital adjustment and thus need something like 3 Bits of extra resolution to keep rounding errors small.
In principle the multislope charge balancing converters found in high end DMMs (e.g. 6,5 Digits) are not that different than sigma delta converters. However the converters are optimized for good linearity using expensive parts (especially resistors arrays) and separate analog and digital chips. Internally the ADC in a 6 digit DMM will also have something like 24-32 bits of resolution.