Based in London, Tannenbaum will oversee new business, marketing, client services and strategy. Her new role will involve working on an agency rebrand and launching an external marketing and events programme.
She will work alongside current MD and co-founder David Beresford.
Tannenbaum's key objectives include further diversifying That Lot's portfolio of clients, developing its offerings into different markets and establishing its position as a "leading creative agency across content and social innovation".
She has been at That Lot for nearly five years, spending the majority of that time as a partner. Prior to her time at That Lot, she worked as a client-side marketer for Bauer Media, News UK and Global Radio
That Lot has grown from five staff to a headcount of more than 90 in the time Tannenbaum has worked there.
"We have been lucky enough to attract some of the industry's most talented creators and social natives, and when you pair our in-house talent with clients with an appetite for innovation and creative excellence, amazing work happens," she said.
"We continue to push creativity and innovation through content that's highly shareable and full of personality. It's a really exciting time of growth for the agency as we take on more global clients and build out our creative, strategic and paid offering beyond London."
That Lot was founded in 2014 by Beresford and David Schneider – an actor, writer and comedian famed for his roles in TV series such as The Day Today and I'm Alan Partridge – along with writer and social-content specialist David Levin.
It specialises in platform-specific social-content production, strategy and management with clients including Channel 4, HSBC, B&Q, Marks & Spencer, Red Bull, InterContinental Hotels Group and McCain.
It was acquired by Weber Shandwick in 2018, making its London operation a network hub for Weber Shandwick's digital offering.
"Weber Shandwick's collaboration with That Lot delivers highly creative, engaging integrated social campaigns for our clients," Weber Shandwick UK and Ireland chief executive Rachel Friend said. "Laura has been a major factor in the success of our partnership, and we’re looking forward to seeing what the team comes up with next under her direction."