Yes, I believe this could and will affect relationships with clients who are very invested in these causes, but it should be kept in mind that a wider distancing from DEI is taking place in corporate America following the election results. Before anything started happening in the legal industry, the biggest players in the tech like Meta and Amazon had already started rolling back on DEI policies.
As for your last point, in Paul, Weiss' case connections to the Democratic party could definitely be a reason they were targeted, as Brad Karp (the firm's chairman) orgainsed a fundraiser for Kamala Harris' campaign, one of the firm's partners coached her for the presidential debates, and other partners were associated with the Robert Muller investigations and claims related to the January 6th events. The same probably applies to Willkie - as I mentioned previously, Kamala Harris' husband is a partner at the firm. However, I would not infer from that that the firms with no strong ties to the Democrats have no reason for concern.
The White House published a list of 20 firms that were being investigated for their employment practices: Debevoise, Cooley, Goodwin Procter, Kirkland, Latham, McDermott, Milbank, Morgan Lewis, Morrison & Foerster, Perkins Coie, Reed Smith, Ropes & Gray, Sidley Austin, Simpson Thacher, Skadden, White & Case, Freshfields, Hogan Lovells, A&O Shearman, and WilmerHale. Some of the firms on this list who are already reportedly looking to strike deals with the Administration, such as Kirkland, are not thought to be particularly close to the Democratic party.