In general, we usually say, "nothing is impossible." That doesn't mean it is easy. There is no built-in function to create a triangular chart of this type. But, if you did all of the math to calculate all of the coordinate points, I imagine you could come close.
This definitely looks like a very challenging task. I'm not familiar with Piper diagrams, but based on some quick google searches it seems that the basic diagram always appears similar to the image you have posted. Values (for example from different water samples) are then plotted on the underlying image. With this in mind, I'm wondering if it would be possible to treat this as a map with a custom background image.
The image would be the underlying structure of the two triangles and diamond shape. The image could even include the symbols, numbers, and grid lines.
A table could be created for the points on each of the triangle plot intersections (for example what are the x and y coordinates of the image where 60% HCO3, 10% SO4, and 10% CI intersect). This could take some time but is possible with time and patience. The more intersections you "map" the more accurate it will be. The intersections at 5% or 10% intervals would be easier than doing every 1% intersection, so how exact you need the diagram to be would become a factor in the time it would take to do this.
My understanding is that the diamond shaped portion is a matrix transformation of the other two. I've never tried to do a matrix transformation in Tableau but it seems like it would be more of a challenge to code correctly. Integrating with R might be useful here to handle the calculations. You would then need to a table for those intersections as well to be able to locate the proper location on the image to place a mark.
Again, this not being an area I'm familiar with, if another diagram such as a Stiff diagram would work instead, that might be easier to create in Tableau. That diagram appears to resemble a radar or spider chart, which I've seen examples of how to create in Tableau using a similar approach to what I described with the use of a custom background image and then mapping points to the image.