Not sure if you are asking about PCI or PCIe. You used both terms, and the answer is different for each.
If you are talking about a legacy PCI bus: The answer is "yes". Board to board DMA is doable. Video capture boards may DMA video frames directly into your graphics card memory for example.
In your example, the video card could DMA directly to a storage device. However, the data would be quite "raw". Your NIC would have no concept of a filesystem for example. You also need to make sure you can program the NIC's DMA engine to sit within the confines of your SATA controller's registers. You don't want to walk off the end of the BAR!
If you are talking about a modern PCIe bus: The answer is "typically no, but it depends". Peer-to-peer bus transactions are a funny thing in the PCI Express Spec. Root complex devices are not required to support it.
In my testing, peer-to-peer DMA will work, if your devices are behind a PCIe switch (not directly plugged into the motherboard). However, if your devices are connected directly to the chipset (Root Complex), peer-to-peer DMA will not work, except in some special cases. The most notable special case would be the video capture example I mentioned earlier. The special cases are mentioned in the chipset datasheets.
We have tested the peer-to-peer PCIe DMA with a few different Intel and AMD chipsets and found consistent behavior. Have not tested the most recent generations of chipsets though. (We have discussed the lack of peer-to-peer PCIe DMA support with Intel, not sure if our feedback has had any impact on their Engineering dept.)