I think neither of these responses fit perfectly into any of the standard GLM link functions. Taking a pragmatic approach it is probably sufficient to pick a link function that is broadly doing the right thing.

Your raw data is from a multinomial distribution, and Richness, $R$, is the number of categories with a score of 1 or more. Although this is count data, it isn't quite the same as the normal sort of count data. The most striking difference being that you can never have a count of zero. As pragmatic first attempt, I would consider modeling $(R-1)$ using a poisson link function. The handwaving reasoning here could be this: If you were catching animals at a random rate for a fixed length of time, the number caught would be modelled well by Poisson. You are catching a fixed number of animals, but at a random rate the type of animal changes. If it wasn't for the fact your return to species already seen, this would be quite a good match.

For diversity, things are even less intuitive. However, the diversity is a sort of average of the abundance ($p_i$) of each species, and as such is bounded by $[0,1]$. This suggests that you could probably treat it as a proportion and use a logit link function.

Advanced Alternative

If the pragmatic matching of support and "looks approximately right" approach is not satisfying, it appears there is another way. I don't know much about this, but there is something called the *Multinomial Diversity Model" which is designed to deal with these sorts of problems. It is described in this paper, http://www.ncbi.nlm.nih.gov/pubmed/23185889 but I have no idea how popular it is.

It is, however, implented in R and availble on CRAN as MRM, apparently by the author of the above paper. It includes a spider data set used as an example in the help:

library(MDM)
data(spider6)
fit0 <- mdm(y2p(spider6[,1:6])~1,data=spider6)
fit1 <- mdm(y2p(spider6[,1:6])~Water,data=spider6)
fit2 <- mdm(y2p(spider6[,1:6])~Water+Herbs,data=spider6)
fit3 <- mdm(y2p(spider6[,1:6])~Site,data=spider6,alpha=TRUE)
anova(fit0,fit1,fit2,fit3)