Can a PIC32MZ SPI Have Multiple Ports Transmitting at the same time?
Yes. According to the "PIC32MZ Embedded Connectivity (EC) Family" datasheet, page 1, it has "Six 4-wire SPI modules". Those peripheral interfaces will be independent.

It also has DMA, so those ports can be read with minimal CPU overhead.

How many SPI interfaces are actually available depends on the package, and which pins are available.