I believe you are thinking of a 'nonparametric bootstrap'. In practice, it works for data from almost all distributions. The fundamental idea is that the empirical cumulative distribution function of the sample is used as a substitute for the cumulative distribution function of the population.
If the data are binary (say, values 0 and 1), then a binomial test is more useful than a bootstrap. If the data are substantially skewed (not essentially symmetrical) then certain 'naive' bootstrap procedures need bias correction to give a useful confidence interval for the population mean. The actual coverage probability
of a 95% CI based on a small sample may be a little less than 95%.
If you have a very large sample (maybe over 100 observations), then
it is probably OK to use a standard one-sample t test even if the
data are not exactly normal. You might compare results from the
bootstrap and t procedures. If they are substantially the same
for practical purposes, then fine. If not, maybe you can try to understand the reason for the discrepancy. (Extreme
skewness, lots of straggling far outliers in one or both tails,
marked bimodality, and so on.)
Recently, I posted an example of a nonparametric bootstrap using
the 'quantile method'. That should work for the type of data you
describe. See question 1577585.
If you want to describe and show your data and the CI resulting from such a procedure, please edit these into your Problem as an
'addendum', and leave me a note beneath this Answer. I will have a look at it.
