Of course you can ! They probably meant the best time was between October and May, off moonsoon season, but there are departures in mid September as well with organised groups, and even before for independant treckers (private departures). Here's an example of one of the dozens of trecking agencies that offer their services for the trip. Everest base camp
. I don't know this agency, I'm not recommending it or not, I found it on google to illustrate my words.
I spent four months in Nepal. Loved it !
My best recommendation would be to choose agencies out of a well known guide books, such as Lonely Planet. You'll understand once in Kathmandu, how difficult it is to choose. There is literaly one agency every 10 meters ! Plus agency's representatives walk on you in the street all the time. Guide books suggest agencies that might be more expensive, but they are 100% reliable. And take the time to shop around for the best price, bargaining is possible. Get an agency that treats its employees decently, ask their salary. Porters and guides are very often (very!) exploited.
Do not book on Internet, it is waaaaaay too expensive. Go to Nepal, and shop your treck.
My personnal suggestion is to take a few days to visit Kathmandu and the surroundings, before or after your treck. Most visitors spend one or two days, and see nothing but Durbar Square and the Monkey Temple. This area is very rich in culture and interests, lots to do and see. Enough for at least a good week !
Trecking gears are extremely cheap there, so you don't need to bring everything from home. You'll be travelling a long time before getting to Nepal, so that could be a good option. The only thing you need to buy before are the boots, in Nepal they aren't of good quality. To give you an idea of prices (from 2004), I met a guy who bought the jacket, gloves, hat, and sleeping bag for $50.
About transportation, there are daily buses and trains entering Nepal at three different border points (east, central, west nepal), from all major northern cities. The border leading to Kathmandu is called Raxaul Bazaar/ Birganj, connecting from Kolkotta or Patna. There are also budget flights at $70-265, depending on the departure city.
Don't hesitate if you have more questions, it's always a pleasure to talk about Nepal !
You're going in September, it should be ab-so-lu-te-ly gorgeous !