Google was the top search provider in the US in December with an estimated 5.42 billion search queries conducted at Google Search, according to a study by Nielsen Online. Google Search conducted 62.9 percent of search queries during December. Yahoo! 
Here is a good presentation that highlights search data for CY 2008
The exact number, if it is possible to obtain it, is probably not related in any sense to you business plan. You probably have to conduct an estimation on other factors in more specific niche or domain.
That is a pretty broad question ;) Do you consider search queries as questions? Do you consider questions asked over email, IRC, IM, Twitter, etc. as questions?
One way to get a very approximative estimate would be to search Twitter for the ? question mark and retrieve the number of results for 24 hours:
http://twitter.com/#search?q=? Knowing that Twitter represents x% of web traffic, you could then estimate the number of questions over the entire web with a cross multiplication.
Another way would be to get that data from the biggest Q&A sites:
Guesstimation (Guess + Estimation) is a technique where you make sensible, smart guess based on the factual data.
It is very useful when one doesn't have all the facts available.
One example could be: How many liters of water do all the seas and oceans hold in them?
The process to arrive at the answer could be as below:
The factual data available with us is:
1. 70% of earth's surface is covered by water
2. radius of earth is approx. 6000 km (i.e. 6000 * 1000 in meters)
3. average depth of oceans or seas is 2 km. (i.e. 2 * 1000 meteres)
(We must convert Kilometers into meters because we want the final answer in Litres)
Now coming to some calculations:
Surface Area of a sphere = 4 x pi x r x r
Thus, surface area of earth => 4 x pi x 6000 x 1000 x 6000 x 1000
Out of this, 70% is covered by water => 70 / 100 x 4 x pi x 6000 x 1000 x 6000 x 1000
Total amount of water considering average depth => 70/100 x 4 x pi x 6000 x 1000 x 6000 x 1000 x 2000 litres
In guesstimation techniques, you won't get the exact number but it said that you can come pretty close to the actual numbers based on the 'smartness' of the guess part.
There's a classical question asked 'how many piano tuners are there in a given city?'
A search for the term 'guesstimation' on google can give you further pointers.