Unfortunately, no. At least, not that way. The BCLK (or SCK) and LRCK (or WS) are inputs on both of these devices (they are both I2S slaves). On the schematic you shown, there is no device driving those signals. It could not work without at least a proper I2S clock generator. You need a clock that drives LRCK at the sample rate you want, and BCLK at 64 times the sample rate you want.
However, you could do this without a microcontroller: grab an oscillator running at 64x the sample rate to drive BCLK (there are off-the-shelves oscillators for this), and a frequency divider that divide this clock by 64 (you could simply use a 6 - or more - bits counter: e.g. 74HC393) to drive LRCK. A microcontroller is actually overkill to generate such a simple clocking scheme.
Anyway, now it can start working. There are a few things to take care of, though:
- INMP441 works in I2S mode only (MSB first, with a 1clk delay between the MSB of each word and the LRCLK transition), with a 32 bit word length. It can output its stream on the left or rigth channel depending on the L/R input, but that's pretty much all you can configure. It will drive the data out line for the L or R word part depending on its configuration, but the data line isn't driven for the remaining part (so you can wire two mics on the same lines to make it stereo). Therefore, you need a pull down on the data line.
- MAX98357x works in a number of modes (left-justified, I2S, TDM). You want I2S. The choice between TDM and the other modes is automatic because in TDM, there is just one pulse on the LRCK line and the chip detects it. The chip won't select this mode, which is fine. Now, the choice between I2S and left-justified depends on the chip variant: MAX98357A/MAX98357B. You want the MAX98357A.
- MAX98357A will aceept 32 bits word length. Fine.
- MAX98357A will take either the left channel or the right channel as input depending on the SD_MODE pin configuration (table 5). Make it consistent with what you decided for the INMP441 L/R configuration.
- INMP441 seem to support any sampling frequency between ~7.8khz and 50kHz (datasheet says SCK should be between 0.5MHz and 3.2MHz). MAX98357A supports only 8kHz, 16kHz, 32kHz, 44.1kHz, 48kHz, 88.2kHz and 96kHz sampling frequencies. Choose something that fits.