There are several ways you could tackle this. There is a package kriging that does ordinary kriging. There are several methods in the package gstat, package raster has contour function for rasters as well as many other things. The package in itself is very complete. The fields package has a Tps function which could satisfy you:
Fits a thin plate spline surface to irregularly spaced data. The smoothing parameter is chosen by generalized cross-validation. The assumed model is additive Y = f(X) +e where f(X) is a d dimensional surface. This function also works for just a single dimension and is a special case of a spatial process estimate (Kriging). A "fast" version of this function uses a compactly supported Wendland covariance and computes the estimate for a fixed smoothing parameter.
Even though it does not seem to be exactly what you want at first glance (I have not used it), it could maybe replace it.
In regards to the other topic, using shiny sounds very cool but I think it could be cumbersome. Not sure. I know it is easy to use but still, maybe it would be easier to use for example the plotKML package and work directly on google earth. I have seen some really beautiful examples. It is very versatile and has a lot of options for aesthetics. It has a lot of interactive options too, like scroll bars to see the evolution of things in time and etc...
This video has awesome examples of what can be achieved with this package. And I beleive it's rather easy to embed google earth into any web page. This is an example of how to embed your earth engine projects into your website.