I've been on this call (both sides of it) probably a dozen times by now. Gov agencies are decent at doing research so it's pretty unlikely that the FBI just called their 1800 number or whatever.
Most small start ups don't get to the level where anyone that "big" is looking at them but in the event that something does get flagged the agency will go find their CEO/CTO/counsel on LinkedIn and either message them there or email them. I've never seen an actual vulnerability disclosed in email, if it's a potential legal issue (hello SEC and fintech) they may ask that your lawyer responds to them in writing but more often it's just "this is Agent XYZ with ABC. I have information about your company, please call me immediately."
For someone bigger (like Citrix) the company is hopefully big enough to have a team that is connected to the agencies in someway. Either the agency knows someone who knows them, or they have a designated Security and Compliance team that can handle these inquires.
The real problems come when you're in the middle of sizes - too big to have eyes on every email but too small to have a real security team.
About 5 years I was working for a SaaS company and one of our clients accidentally discovered a pretty serious hole in another company's product. This client wasn't overly tech savy and was basically like "hey is this how this is supposed to work?" when it very much was not... so we killed the API connection and told the client we'd take care of it. It's about 7pm ET by the time we figure out what's going on so we call and email the other company but couldn't find anyone. In the end we got the home phone number of their CTO and had our CTO call him at around 10pm. He thought it was a prank call but once our CTO convinced him this was a problem he was able to get their on call eng to patch it within hours.
Nowadays almost any company involved in security work either has a direct line to FBI/DHS or has a vendor who does. ie if I'm some medium consumer platform I probably don't get to talk to the FBI directly, but if I called up Crowdstrike or any security consulting firm they could do that. In the event that my medium consumer platform was infiltrated by Fancy Bear (and the government decided to tell me, sometimes they don't) an FBI agent would email/call the most likely point of contact for the fastest resolution without causing panic. Lots of time the damage is already done, two vs four hours on a response won't make a big difference in the long term so no need to email info@ or anything.
Over the past 6-8 years the corporation on public/private cyber investigations has definitely changed as red tape has decreased in sharing of info has increased - even more the last 4ish years since the DNC email hacks. I've had a clients get a casual "just a heads up, you should check this out" from the government without no paperwork and no follow up, something that would have been virtually unheard of 8 years ago.
DHS gets a lot of shit in the media (lots of which is deserved) but they've done a pretty good job just opening basic lines of communication and training other agencies that spending 20 minutes looking at a random tip, and following up if needed, is actually a pretty good use of time.