Not sure about the Weibull option. However, you might find Tweedie distributions helpful for fitting non-negative, over-dispersed data with a spike at zero. See R packages tweedie and statmod. Not surprisingly, a bit more finicky to use than your typical glm, but I've had some success with twee die in the past. I'm assuming you're not interested in a hurdle / mixture approach, which is another common and well supported approach.

You might want to try wind data, which are sometimes modelled with Weibull, but which seem to be reported as zero whenever they are small. I do not have an exact dataset (or URL) to suggest, because the datasets I'm finding by looking at Environment Canada (for my local winds) are all just giving hourly data on short time periods. Please note that I've not looked into these distributions in any detail, but I have noticed a pattern of zero-excess when using data for other purposes.