The word choice in the question is quite particular, so my answer will reflect this particularity: "vulnerable" is defined as "susceptible to physical or emotional attack or harm," while "susceptible" is defined as "likely or liable to be influenced or harmed by a particular thing."
So really the question is asking whether LinkedIn is competitively likely or liable to be influenced or harmed in any way in any of its core products from startups or other large companies like Facebook.
LinkedIn's "core products" are the trifecta of (1) job search; (2) talent search; and (3) professional networking. To determine whether these core products are potentailly at risk, let's look at LinkedIn as a whole, from a macro level: No, it is not "likely" to be harmed. Possible? Sure. Likely? No.
By these definitions then, no, LinkedIn is not competitively vulnerable from startups or other large companies like Facebook.
Let's address the elephant in the room first: despite what Facebook may want to accomplish -- and what people may think -- no, it's not going to replace LinkedIn any time soon -- or ever -- for the sole reason that people do not want Facebook to be LinkedIn, and they do not want LinkedIn to be Facebook. To wit, people (most of the time, anyway) desire and intend to keep their professional and social lives separate.
So that's Facebook. Now let's address other startups or other large companies generally.
The sole reason a product like LinkedIn -- and for that matter, Facebook -- works is precisely because of its network effects: the more people are a member, the more those members benefit. And once the network effects are firmly rooted, it's highly unlikely for it to be uprooted, as long as the existing customer base is happy enough. There's just too much of a utility cost -- a burden -- of moving from one platform to another.
So in order for an established network effects-reinforced company to be dethroned, two things need to happen contemporaneously: (1) a massively superior new product needs to enter the marketplace; and (2) people need to be seriously unhappy with the existing product.
But as I addressed in my answer to What product changes does LinkedIn need to make to improve growth in 2016?*
the problem with LinkedIn -- or its blessing, depending on your point of view, I suppose -- is that it's just good enough because it gives people hope
. It's neither terrible nor great, it's just sort of there, it's the de facto standard for a professional network, and it's just what you're expected to use in today's professional world.
LinkedIn, then, is like the yellow pages and white pages for the professional world. It may not be optimal -- it may even be suboptimal -- but it's there, it sometimes works more or less, there isn't anything better, and crucially, it always offers that glimmer of hope that maybe, just maybe, it will help you strike it big one day.
So then the question becomes, (1) how likely is it for a vastly superior product to come along and (2) how likely is it for enough of the customer base to become seriously unhappy with the existing product?
While many new products will surely be introduced in an attempt to battle LinkedIn, unless and until a vast majority of LinkedIn's customers decide to leave LinkedIn, then it will be of little consequence. And there mere presence of a (potentially) better product will not suffice; it must be both the presence of that (potentially) better product and a massive discontent with LinkedIn to catalyze the shift.
A good example was MySpace. It certainly enjoyed healthy network effects -- by c.2004 standards, anyway -- but ultimately people fell out of interest with the platform. I can't remember particular problems with it now, but it certainly became spammy, ugly, unpleasant to use, inundated with ads, and just generally became an unpleasant place to be. Then Facebook came out, and little by little rolled out beyond college campuses. Yes moving social networks from MySpace to Facebook was a burden, but enough MySpace users were sufficiently disenchanted by MySpace that the cost/burden of moving to Facebook was worth it.
Facebook prevailed over MySpace not simply because it was a better product, but also because people were fed up with MySpace.
So unless and until LinkedIn manages to sufficiently upset a majority of its users and a legitimately superior product comes along, LinkedIn will remain where it is. Not because it is great, but simply because people like it just enough to stick around now that they are committed to its ecosystem.
* Also in my answer to that question, I went into a detailed discussion about why LinkedIn is simply "good enough," with respect specifically to its core products.