This is something that gets a bit complicated but is ultimately a stylistic point not to get too caught up on. You are not going to be marked down on something like this.
In short, because you are not writing as if you are “the firm”, I wouldn’t capitalise it.
Some firms do capitalise firm when describing themselves though (eg when they publish articles and want to refer to themselves), whilst others don’t. Many firms these days actively avoid writing “the Firm” and instead will use other pronouns to define themselves (we/our mainly), and therefore you can also try this approach to minimise the use of “the firm”. This often saves on word count too. For instance, if you are trying to describe the firm you are applying to, you can always use “your” - “the Firm’s clients” can easily become “your clients”.
However, for those that do capitalise “the Firm”, they are identifying themselves as a particular entity. Firm generally is not capitalised if you wrote “a firm” or “the firms involved in the project”. Legal documents are a whole different ball game though.
This article here explains it better than I can!
Someone smart once said you have to pick your battles. That’s particularly true at Baker McKenzie, where trying to effect change can feel like attempting to shift the direction of the Seawise…
thewritinghabit.blog