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I can create a least cost path (LCP) using a shapefile of points over a "cost surface" raster - But can I do polygons and not points?

This is not the same as the LCP between the closest points of each polygon. You'd have to know the best path from between the polygons "as a whole". See the illustration.enter image description here

I want to find the LCP for a conservation corridor between "cores". Let's say you have a to get from on area to another area, and there's a mountain in between - the best path is around the mountain, and from a certain exit and a certain entrance.

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    I can remember facing a similar problem to locate bioregional corridors in order to identify which areas of native vegetation were (statistically) particularly important and which were too isolated to be significant, in the end I used the brute force method, calculating cost path for each vertex then keep the 'best' one. You'd need to use some python if you were to follow a similar approach. Hopefully someone has found a better way in the 10 or so years since.
    – Michael Stimson
    Commented May 25, 2023 at 3:55
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    Unless there's some specific features relating to your data that you can use to reduce the search space or otherwise optimise, I'd expect that you're going to need to use a brute force approach as indicated by @MichaelStimson.
    – Tom Brennan
    Commented May 25, 2023 at 6:13
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    Even a brute force approach is not that bad really in a case like this, most graph libraries can do 1->n paths search and distance matrix (n->n) or even n->n paths search
    – Kalak
    Commented May 25, 2023 at 6:50

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