The problem is that this construction is trying to look for a "data" folder within your existent "...\Vector Map District Raster\data" folder. Since there is no such "...\data\data" folder, it returns an error.
You'd need to iterate through each folder, but as far as I know there is no way to do this with gdalbuildvrt (or any other GDAL utility, for that matter). You have three options here, as I see it:
1) Put all your TIFFs in a single folder. This way, you can call the utility as:
gdalbuildvrt Vector_district.vrt "folderName\*.tif"
2) Create a list of file paths for each raster, and put it in a text file (one filepath per line), as such:
and so forth. Save this list as a txt, and call it in the utility:
gdalbuildvrt Vector_district.vrt -input_file_list myRasterList.txt
3) Do it in python. For this you'll need the GDAL python bindings, which you can get from here. Then you do:
from osgeo import gdal
os.chdir(r'C:\Users\*****\Desktop\Os Raster\Vector Map District Raster\data')
li_dirs = [folder for folder in os.listdir(os.getcwd()) if os.path.isdir(os.path.join(os.getcwd(), folder))]
li_all_files = list()
for folder in li_dirs:
path = os.path.join(os.getcwd(), folder)
li_files = [file for file in os.listdir(path) if os.path.splitext(file)[-1] == '.tif']
for file in li_files: