Who gathers the production/use data that makes up GDP? And how? I know that GDP can be calculated from the national accounts using data which says how much of each industry's output was bought by each other industry.

But how is this kind of supply/use data gathered? Is it via questionnaire?

If so, is a random sample of businesses quizzed, or does every business have to submit data on what they bought and from whom?

Where is this data-gathering process documented? (For, e.g., the UK.)
In the UK, which uses what is know as "the output approach", the Office for National Statistics (ONS) sends out around 45,000 surveys to business, called the Monthly Business Survey (MBS).

The ONS claims that this covers "around 55% of the economy.".

In this survey, questions about turnover and purchases are asked which then allow the ONS to assign total outputs and intermediate expenditures to industry classifications. In the case of the ONS, they use Standard Industrial Classification (UK SIC 2007).

There is a separate survey for construction called the Construction Output Survey, and there are additional forecasts and models to fill in the gaps.

These output and expenditure estimates are then deflated by things like the Consumer Price Index (CPI) to produce an inflation-adjusted GDP figure.

More detail on this process along with many more links to data sources is available in the GDP quality and measurement information document.