Do Buddhists file lawsuits when grievances are done against them?

This may depend on whether you're a lay follower or a monk.

While violence and destruction is strongly discouraged in Buddhism, it is ok for a bonafide ruler or government ("wheel-turning monarch") to establish courts of justice, police and armed forces to:

• protect and guard the people
• ensure peace
• ensure that justice prevails

As such, I don't see why we shouldn't have civil litigation mechanisms for justice and I don't see why lay followers shouldn't use them.

"What have you heard, Ananda: do the Vajjis duly protect and guard the arahats, so that those who have not come to the realm yet might do so, and those who have already come might live there in peace?"

"I have heard, Lord, that they do."

"So long, Ananda, as this is the case, the growth of the Vajjis is to be expected, not their decline."
DN 16

‘But sire, what are the noble duties of a wheel-turning monarch?’

‘Well then, my dear, relying only on principle—honoring, respecting, and venerating principle, having principle as your flag, banner, and authority — provide just protection and security for your court, troops, aristocrats, vassals, brahmins and householders, people of town and country, ascetics and brahmins, beasts and birds. Do not let injustice prevail in the realm. Pay money to the penniless in the realm.
DN 26

However, monks are expected to be completely pacifist according to SN 35.88, MN 20, MN 86, MN 21.

"Well then, Punna. Now that I have instructed you with a brief instruction, in which country are you going to live?"

"Lord, there is a country called Sunaparanta. I am going to live there."

"Punna, the Sunaparanta people are fierce. They are rough. If they insult and ridicule you, what will you think?"

"If they insult and ridicule me, I will think, 'These Sunaparanta people are civilized, very civilized, in that they don't hit me with their hands.' That is what I will think, O Blessed One. That is what I will think, O One Well-gone."

"But if they hit you with their hands, what will you think?"

"...I will think, 'These Sunaparanta people are civilized, very civilized, in that they don't hit me with a clod.'..."

"But if they hit you with a clod...?"

"...I will think, 'These Sunaparanta people are civilized, very civilized, in that they don't hit me with a stick.'..."

"But if they hit you with a stick...?"

"...I will think, 'These Sunaparanta people are civilized, very civilized, in that they don't hit me with a knife.'..."

"But if they hit you with a knife...?"

"...I will think, 'These Sunaparanta people are civilized, very civilized, in that they don't take my life with a sharp knife.'..."

"But if they take your life with a sharp knife...?"

"If they take my life with a sharp knife, I will think, 'There are disciples of the Blessed One who — horrified, humiliated, and disgusted by the body and by life — have sought for an assassin, but here I have met my assassin without searching for him.' That is what I will think, O Blessed One. That is what I will think, O One Well-gone."

"Good, Punna, very good. Possessing such calm and self-control you are fit to dwell among the Sunaparantans. Now it is time to do as you see fit."
SN 35.88

It depends on your current focus of Buddhist practice.

If you are at the level where you are trying to master controlling your emotions, and the big company's actions trigger powerful emotional response in you, then you would focus on letting go (un-sustaining) of that emotion. If you succeed, then your main problem is solved. At that point you may decide to take legal action or abandon it based solely on rational reasons, which often (but not always!) ends up in the topic getting abandoned, since we the humans are rather emotional type of sentiment beings.

Similarly, on the next level where the main focus of Buddhist practice is your ego, which always has a tendency to exaggerate its importance, posture as the deeply offended party, absolutize its perspective as the only valid one, and so on, - then in context of the practice of uprooting the egocentric habits a Buddhist may meditate on the subject of the conflict between themselves and the Big Organization and try to expand one's perspective as to see the situation from outside of the ego's context. If that ends in an authentic success then more often than not the Buddhist will actually see that his own personal grievance is not really important in the great scheme of things, resulting in the issue getting abandoned.

If the Buddhist has passed both of these levels: learned to control one's emotions, mastered the noble skill of letting go, and achieved perfection in seeing things (and acting!) from outside of one's ego - then comes the time for cultivating the power of intent. On this level, the Buddhist won't have emotions nor personal biases associated with the situation, but may choose to litigate solely for the exercise of courage and will power. This is nice especially if the Buddhist previously had a history of laziness or has a tendency for giving up. Again, it's not done for any objective reasons, but primarily in context of one's main Buddhist practice at the time. It goes without saying that the reason for litigation must be wholesome to begin with. In other words, it has to be objectively examined and established as conducive to peace (both at the personal and the global level).

Finally, if a Buddhist is through with all this training, they would make a decision to act or not to act based solely on this last criterion. Once the personal training is out of the picture, the global peace ends up being the primary decision making factor.