Organisational legitimacy is generally considered to be a dichotomous phenomenon; either you have it, or you don't. However there are a few studies that operationalise legitimacy on a scale. See these studies:
Ruef, M., & Scott, W. R. (1998). A multidimensional model of organizational legitimacy: Hospital survival in changing institutional environments. Administrative Science Quarterly, 43(4), 877-904.
Deephouse, D. L. (1996). Does isomorphism legitimate? Academy of Management Journal, 39(4), 1024-1039.
See also this study:
Vergne, J. P. (2011). Toward a New Measure of Organizational Legitimacy: Method, Validation, and Illustration. Organizational Research Methods, 14(3), 484-502.
if you conceptually broaden the issue and not focus too much on legitimacy, but instead think in terms of e.g. trust, there are several empirical studies of public trust in specific political parties, union representatives trust in top management, etc.
most of these use a simple 1-5 scale from low/no trust to high/complete
but then again, I am somewhat sceptical regarding what people actually mean when they say they trust someone. If you think about politics, I might not like the political ideas of a specific party or their leader, but I trust that they will act in acordance with them, that they won't be corrupt, etc....