In this sentence With is equivalent to “because of”.

"Because of", "given that", "due to", 'on account of the " are all acceptable.

Here "with" is synonymous with "Given that" or "Because". It is followed by a statement of fact which then leads to some other conclusion. You have some errors though. The correct sentence would be: With car sales surging in July by 105 per cent, government officials are claiming that the country on back on the path of rapid recovery. Or you could say, "With the car sale surge in July of 105 per cent ..." or "Given that car sales surged in July by 105 per cent ..." or "Because car sales surged in July by 105 per cent ..." Does that help?

Most welcome