Can an AUTOSAR BSW stack (eg. for CAN Communication) be developed based on the specifications provided on their website without purchasing any of the expensive vendor tool? What would be the steps that can be followed? I have been asked to explore this possibility.
Yes and no. The tooling is a necessary part of the equation in AUTOSAR.
Assuming you're working for an automotive supplier, you'll get ECU-specific set of config files (ECU extract?) that is vaguely similar to, but far more extensive than various exchange formats for CAN messaging. These are typically a bunch of XML files of a format defined by the AUTOSAR consortium. A fairly complex chain of "expensive vendor tools" then converts this into something that can generate code. All the specs are open so you could write your own tools in theory.
On top of the tools, you also need the OS, which could possibly built on top of an open source OSEK system as a starting point. Then there's the MCAL.
Can it be done... Yes. If you work for a huge company (e.g. Bosch) it might be in long-term in your best interest to develop your own solution. But if you work for a smaller scale supplier with a project deadline, you'll probably have to bite the bullet and write the massive check. Hopefully this was factored in when your company bid on a job for an AUTOSAR-based system.
To give you an idea of the scale of such an undertaking, it would be a bit like saying, "I'm writing a cool game, do I need to use Windows, OSX or Linux, or can I write my own operating system too?". Ok, maybe not quite that extreme, but significantly harder than writing your own RTOS and vehicle bus stacks (CAN, LIN, etc). If writing your own RTOS + CAN stack sounds prohibitively complex/difficult, don't even think about it.
Obviously I cannot make the decision for you, but if you're taking opinions from random people on the internet of unstated qualifications, I'd highly recommend against it as a "my first autosar" project, but keep it on the table as a possible internal R&D project with possible long-term savings if you're at a top 50+ automotive supplier doing multiple autosar projects every year. I doubt you could build it for less than it would cost to license everything once. But maybe 5-10 big projects down the road it might pay off.