OK I've checked the numbers they listed in the main article and they do match up with the spreadsheet data I have.
However I would take the percentages they show with a big pinch of salt, because they've used ONS population estimates to calculate them, rather than the additional population data provided in the spreadsheet which comes from the national immunisation management system. This can make a fair difference, eg for my area they have 88.8% on that website, but using NIMS population estimates rather than ONS ones, I get a percentage more like 83.8%.
Its a shame the site doesnt do MSOA level of detail in this regard properly, because drilling down to that level of locality is likely to give people a better sense of how things are going for them locally. Quite a lot of important detail and clues are lost if we stick only to larger local authority areas, masking much detail when it comes to BAME populations, poverty etc, but local knowledge is required to get a sense of this.
Likely I will mess around with the raw spreadsheet data in the coming days and will be in a position to share MSOA-level data with people who are interested. But since there appear to be 6,791 MSOAs listed for England, I will probably have to take requests. For example if someone requests Lambeth, there are 35 MSOAs in that area that I can calculate the details for.