If you can find canning or pickling salt, it's not iodized (and has no other additives, unlike some kosher salts), so won't contribute any off flavors that roux mentioned.
However, the different grain sizes will be a problem as they pack denser, so based on a chart from Marton's salt that ManiaxZX linked to in discussing differences in salts, they'd recommend cutting the salt by 20% when changing from kosher salt to table salt to canning/pickling salt.
Saltpeter is potassium nitrate, which does not directly cure meats. Bacteria convert nitrate into nitrite, which is the real preservative. Saltpeter can be replaced by a smaller amount of nitrite to get the same curing effect (most commercial cured meats do this), though a prolonged cure that converts nitrate into nitrite can develop more flavor.
Tender Quick is not a direct substitute because it contains mostly salt. I've heard that you can replace the salt in your recipe with Tender Quick, and drop the saltpeter, and have a success. You would have better luck finding a recipe that was meant to use Tender Quick, though.
It is definitely possible to buy (food-grade!) saltpeter. I would check online, or at specialty stores. It's a little more difficult than picking it up at your local grocery store, of course.
(Chemistry lesson, courtesy of McGee: nitrate (NO3) is converted to nitrite (NO2), which then reacts to form nitric oxide (NO), which bonds to myoglobin in the meat, which turns it pink and prevents oxidation. Nitric oxide is also present in smoke, which gives that "pink ring" around the outside of smoked meats.)
Best Answer
No, not really. There are plenty of curing recipes for pastrami that do not use curing salts and you can use Himalayan for them. It is really just rock salt with some iron oxide (rust) in it. Some people claim an improved taste, others say it is imaginary, but it is a substitute for table salt, not curing salt.
As said, plenty of recipes do not use curing salt, and you can make a pastrami you may really like. But the more vivid pink color will not be there. That color comes specifically from the reaction with the curing salt nitrates and/or nitrites, not the pink coloring added to many of the salts. That coloring is added as a warning that this is not table salt, don't confuse it and use it as table salt. The current popularity of Himalayan is confusing some people and could cause accidents that need to be avoided.