Can we start acquiring links for a site within a week of its launch?
We tried doing this for a previous site and got penalized by Google. How can we acquire links fast, legitimately without getting penalized (e.g. asking other high ranking sites we have existing relationships to backlink to us)
It's not abnormal for a quality site launch to have press releases and press coverage. Perhaps your past method featured too many borderline spammy links--enough to emulate the types of sites that Google has trained its algorithm to ban.
I'd challenge you to think of your site launch as a new product for a well-established brand, or a new and exciting startup. What kinds of sites do you usually see covering those types of launches? If your site isn't doesn't carry the excitement of one of these types of launches, your challenge is to be creative & find the right angle. I have yet to find a niche that I couldn't create some sort of interesting hook. For example, insurance sounds incredibly boring to me, but tell me the stories of people wrecking $100k cars or interesting data on the most dangerous cities to drive in, then you've got me hooked.
Every SEO gets links in different ways, but my preference is to shoot for the top and get coverage in the publication that other news or blog outlets tend to cannibalize. I'd rather gain 100 backlinks that way than to reach out to 5,000 sites in hopes that 100 of them will do some sort of link exchange that can sometimes hurt you more than help you.