Sounds like STEMNET is what you're looking for, at least for some of it

It seems to me that my company (Arcadis) is starting to get involved in the STEM activities.
Feel free to contact me.

Definitely a challenge! You also might want to look for people with titles that include Outreach and Social Sustainability. Also reach out to people who are the Communications team for companies you think are worth approaching.

You will find that any company working on a government funded project will have a social sustainability target to meet. However, there may be a tight geographic boundary to that requirement (i.e, if the project is in Aberdeen, they probably won't be interested in schools in Cornwall).

My company (like many others) offers 2 days per year for me to use for volunteering - which I utilise as a STEM Ambassador. However, I rely on schools making requests through the STEM Ambassadors system.

100% agree with all Mark's points.

I'd also add that in my experience (having been very, very heavily involved in STEMNET and other organisations in the past as a volunteer) the idea of "getting companies interested" is flawed. "Companies" aren't going to be interested, individuals are. Those individuals may be rank-and-file engineers who either give up their own time or (as Mark says) work for companies who give them some time off for volunteering. But, if you're very lucky, they may be senior managers or even the owners of a company, in which case they may volunteer several of their staff. However it's important to remember that it's still down to finding those individuals, and you'll be lucky if you find them through the company infrastructure. (Never to say never, just not to pin too much hope on it.) So when it appears that a "company" is interested, what I've found it means is that a senior manager / owner of that company is personally interested.

I'd actually say from my experience that it's typically much easier for schools themselves to find these people, through parents, friends of parents or teachers, etc. Next most likely is that a charitable association will find them through similar means. However, for a commercial operation (however small) to find volunteers to help with their activities (however worthwhile) is always going to be a challenge. If someone is running a small (or indeed large) business and they hear that their child's or friends child's school would like some help then that's a potentially powerful personal draw. Ditto with a similar charitable relationship. However as soon as it's a business asking them for help then it becomes a B2B relationship - which is a much harder sell.

I can give a few pointers to how to do that "sell" (being someone who has long been on the receiving end of these requests!):

• DO NOT ASK FOR MONEY, unless you are approaching a very small business. It is incredibly difficult for a medium or large company to give money to a non-business related activity (excepting where staff themselves organise fundraising, but even then they will want it to go directly to the activity, not to the organisers). It doesn't matter that you are only asking for a small/tiny percentage of their turnover, usually there is simply no mechanism for giving random people money - not surprisingly and quite reasonably!
• Unless you know them well, don't ask straight out for materials, for the same reasons. And if you do, don't be greedy, be grateful.
• Do ask for employees' time. This is the most likely thing businesses can provide. And the sell here is that particularly for apprentice or recent graduate staff it can be really valuable training.
• Offer press coverage. Some will want it, some won't. If the business sells locally they'll be interested, or possibly if they recruit locally. But be aware that many engineering companies aren't that interested in local press coverage (unless they're about to e.g. apply for planning permission!), so make sure where possible to show that they can get national coverage in places their customers may see - even if it's just you ensuring they have good photos for their internal PR departments. This means being professional about ensuring they can get decent quality and fully cleared photos, perhaps with their company logo showing, and definitely an indication of the activity. But equally respect the fact that some won't want publicity (perhaps because they don't want to be plagued with requests). 
• Odd one which often seems to get forgotten: schools plan activities a year in advance, companies often plan activities weeks or days in advance and then change their plans anyway. Don't expect engineers or managers to be able to 100% commit to something six or nine or twelve months away.
• Do actually let the engineers who are volunteering bring and use their own expertise. It is very dispiriting to give up a day to be basically doing crowd control (been there, done that). Whereas actually engineering with the students is great fun and a fantastic learning experience. So this needs to be very clear in the "sell", and it must be stuck to. To slightly oversimplify, I very happily volunteer to stack chairs and brew the tea for charities or similar, I don't expect to be asked to do it for a commercial concern!
• Did I mention not to ask for money? 

I still feel as passionately as I ever did that we need to get more working scientists and engineers into schools. But (apologies, micro-rant coming) what hugely frustrated me in the last few years when I was doing a lot of STEM volunteering was that too many of the organisations involved seemed to be forgetting that this is for the mutual benefit of the students and the volunteers, it is not for the benefit of those organising bodies. The way to attract volunteers and corporate support is to give a clear focus on the students and volunteers needs, and to make sure this is absolutely carried through.

Thanks - and Brian, best wishes for a successful campaign if you're still following this thread, hopefully something in the above will help a bit.

Andy