We live in an affluent, highly educated area. Our schools are in the top 5% in california.
Our district tests all third-graders with the OLSAT, but they will also accept some private testing with the WISC and a few other IQ tests. To qualify for our gifted program, your child must get at least a 96th percentile total plus 96th percentile on either verbal or nonverbal.
I recently learned that 15% of the third-graders in our district got at least one 99th percentile on some part of the OLSAT or a private IQ test. Does that seem too high??

Without doing detailed statistical calculations - not at all. You've got two effects going on: the bell curve in your area isn't the standard one, and multiple measures of which we're only looking for a 99th percentile in one. Thinking about what I know of highly educated bits of California, I guess the first effect on its own might account for almost all of this.

I think this is not implausible. Let's make a few coarse-grained assumptions and run some numbers:

Given the demographics of your district, let's say the top 30% are equivalent to the top 5% of the general population. (That could be because of intelligence leading to affluence; enriched early experiences leading to better brain development; or plain old hothousing.) Of those, 1/5 will actually be in the top 1%. What about the others?

Remember that there will always be measurement error. That puts those kids within range of scoring in the top 1% by accident. Most of them won't (they are just as likely to under-perform as to over-perform). But that's just on a single measurement. They get at least two shots (the two subtests of the OLSAT) and possibly more -- possibly several more -- if the parents go for private testing.

I could easily see something like half of those who are "really" in the top 5% managing to get at least one score in the top 1%.

What you say makes total sense. We are in a district sort of like Palo Alto, where Stanford is.
still, wow! They are always trying to shrink or even get rid of our gifted program since it is a full-time self-inclusion program. I feel like they should expand it, not shrink it, since there is always a big waiting list.

Does not surprise me at all. My DDs have changed schools recently and at their old school I am quite sure that there would have been at least 15% of the school in the gifted range and at least one HG child per grade (which was 1 in 50 children). There are definitely still a higher than average number of bright children at the new school, but not nearly so many gifted and I certain far fever HG/+ children, the difference is marked.

There is an organized group of parents who are trying to close down our local public school gifted program! It's so sad. I'm not sure what to do.

The OLSAT is a school readiness exam and not a gifted testing instrument. Scores on this and on IQ tests tend to correlate for those in the intellectually disabled category and for typical students but do not correlate for gifted and especially do not for highly gifted. Highly gifted children do pretty poorly given their skill set on this exam. My son had a FQIQ on WISCIV as 145 and got the 90th percentile on verbal on the OLSAT.

I can see where as many as 15% in affluent districts in 3rd grade get above 95% (possibly 99%) as it is not geared to find gifted but more to find kids who will do well in school. The questions for kindergarten are like 4 oranges objects and one yellow and the student is asked which one doesn't belong. Smart kids will answer the yellow one, but gifted kids might not. My son for example chose the sun. His reasoning was because it was the only one that did not touch the ground. Affluent areas where many are given lots of prep for school should scored above and beyond on these type of school readiness exams. But that does not mean that the kids are gifted.