Not unless you're using multilevel modeling to account for this. Just repkicating onservations within organization and calling them different is invalid.
If you are interested in a quantitative survey, I think it depends upon what kind of information you are collecting, but it sounds like you might want to research "multistage" "cluster sampling." I mean, you'd research the terms "multistage sampling," and "cluster sampling," because you might want to do a two-stage cluster sample, where you first select organizations, perhaps with a simple random design, and then within each organization (cluster) selected, you could do a simple random sample of respondents. Or you could make this a one-stage cluster sample by doing a census of respondents at the organizations selected. Organizations might be drawn with probability proportional to size (number of respondents possible). More complex two-stage cluster designs could also be done. You could consult a textbook such as W. G. Cochran(1977), Sampling Techniques, 3rd edition, Wiley. Also, I think Statistics Canada has long had software that will do many variations on this.
If you are doing a qualitative survey, I suppose the principle is the same, except that I don't know how you would sample within a cluster (organization) since you then do not know if you are over- or under-representing one cluster (organization) compared to another. Perhaps you would select the same percent of people from each organization? I don't know.
One- or two-stage cluster sampling might be good, particularly if you are collecting continuous data at the respondent level, but if each respondent is answering for an entire organization, and the difference between respondents for the same organization is just measurement error, then more observations within an organization might give a better result for that organization, but that's not the same thing. You would then want to know if the goal is to just know something about those particular organizations, or do you want a sample of organizations to infer something for all organizations.
If you are sampling from all organizations, so that you may infer to the population of organizations, for organization level data, then the number of respondents from an organization just depends upon some kind of measurement error. I think this is probably what Matt had in mind. A paper that Kelvyn Jones showed me that he has work on for multilevel modeling notes using it to cover measurement error, though I do not know if that applies to you. For one thing, you'd need predictor variables. Here you are more likely to just see what range and standard error for a mean response is found when collecting more than one response from an organization, if it is an organization level response. That would mean doing some experimenting, perhaps in a pilot survey.
Regarding sampling in general, you probably need a randomized design. Stratification, say stratified random sampling, is often very good to help obtain better overall results, but good results for each stratum is a different problem.
The following might be of interest, regarding inference from sampling in general:
Brewer, K.R.W. (2014), “Three controversies in the history of survey sampling,” Survey Methodology, (December 2013/January 2014), Vol 39, No 2, pp. 249-262. Statistics Canada, Catalogue No. 12-001-X. http://www.statcan.gc.ca/pub/12-001-x/2013002/article/11883-eng.htm
I agree with James R Knaub. In addition, collecting data from multiple respondents in each organization can be used as a one of the ways to overcome common method variance given that each of the respondent provide information on different variables. For instance, one respondent may provide info on the dependent variables while another may provide info on the independent variables.
Put simply if you just have one individual from each organisation then two sources of variation are confounded and cannot be separated: between person within organisation , and between organisation get conflated into between person/organization.