Almost certainly not.
For many reasons, it’s incredibly unlikely that Umbridge would have fought for either side at the battle at Hogwarts. She wouldn’t have wanted to fight for the Order since they weren’t in favor of any of the policies she’d want, and she wouldn’t want to fight for the Death Eaters because they’re clearly an outside-the-law organization, and she prefers hiding behind the might of the Ministry. In addition, she wouldn’t want to join a battle on any side, she doesn’t have the temperament.
Umbridge liked to hide behind the law.
Umbridge did everything she’d done hiding behind the law and the Ministry of Magic to justify her actions. J.K. Rowling describes her as doing what she did in the name of law and order, contrasting with the Dark Lord being ‘openly evil’.
“Her desire to control, to punish and to inflict pain, all in the name of law and order, are, I think, every bit as reprehensible as Lord Voldemort’s unvarnished espousal of evil.”
- Dolores Umbridge (Pottermore)
If she fought in the battle along with the Death Eaters, this wouldn’t allow her the pretense of law.
Umbridge never declared allegiance to the Death Eaters.
Though Umbridge was known to be ‘a nasty piece of work’, she’d always remained with her allegiance being to the Ministry, not the Death Eaters. Sirius tells Harry that Umbridge is horrible, but definitely not an actual Death Eater.
“I doubt it,’ said Sirius. ‘I know her by reputation and I’m sure she’s no Death Eater –’
‘She’s foul enough to be one,’ said Harry darkly, and Ron and Hermione nodded vigorously in agreement.
‘Yes, but the world isn’t split into good people and Death Eaters,’ said Sirius with a wry smile. ‘I know she’s a nasty piece of work, though – you should hear Remus talk about her.”
- Harry Potter and the Order of the Phoenix, Chapter 14 (Percy and Padfoot)
Even when she got more authority under the Dark Lord’s control of the Ministry, it wasn’t because she had become a Death Eater or had become willing to openly take their side- the Death Eaters simply judged that she’d be willing to enact the laws they wanted. She remained with the Ministry.
“Correctly judged, by senior Death Eaters, to have much more in common with them than she ever had with Albus Dumbledore, she not only retained her post but was given extra authority, becoming Head of the Muggle-born Registration Commission, which was in effect a kangaroo court that imprisoned all Muggle-borns on the basis that they had ‘stolen’ their wands and their magic.
- Dolores Umbridge (Pottermore)
Going into battle for the side of the Death Eaters would be tantamount to declaring her allegiance to the Death Eaters, which she wouldn’t want to do. They’re ‘outside the law’, whereas she could stay in the Ministry and still try to portray herself as acting within the law, whichever side won.
Umbridge was too afraid and incapable to willingly go into battle.
In the only somewhat battle-like situation we see Umbridge in, she doesn’t enter it willingly, and doesn’t do well once in it. She’s tricked into going to the Forbidden Forest, and when she’s confronted by the centaurs, she’s absolutely terrified. While she does fear half-breeds, so would be especially afraid of centaurs, it seems unlikely she’d respond well to a chaotic battle, and one where half-breeds were participants and fighting for both sides.
“An arrow flew through the air and landed with a menacing thud in the tree just over her head. The air was suddenly full of the sound of hooves; Harry could feel the Forest floor trembling; Umbridge gave a little scream and pushed him in front of her like a shield –”
- Harry Potter and the Order of the Phoenix, Chapter 33 (Fight and Flight)
Though she raised her wand in a feeble attempt to defend herself, she’s very shaky and clearly panicked. In addition, rather than actually use her wand, her first strategy is to attempt to convince the centaurs that it’d be illegal to attack her and the Ministry would prosecute them if they tried.
“Harry looked left. The chestnut-bodied centaur called Magorian was walking towards them out of the circle: his bow, like those of the others, was raised. On Harry’s right, Umbridge was still whimpering, her wand trembling violently as she pointed it at the advancing centaur.”
- Harry Potter and the Order of the Phoenix, Chapter 33 (Fight and Flight)
When she finds her attempts futile, she does attempt to fight back, but only is able to fire one spell, but mainly reacts by screaming in panic.
“Harry saw a flash of red light and knew she had attempted to Stun one of them; then she screamed very loudly. Lifting his head a few inches, Harry saw that Umbridge had been seized from behind by Bane and lifted high into the air, wriggling and yelling with fright.”
- Harry Potter and the Order of the Phoenix, Chapter 33 (Fight and Flight)
She doesn’t seem to be a capable duelist, and doesn’t have the temperament that would make her either suited or willing to join in the battle. She’d much rather hide until it’s over.
Umbridge craved control.
Umbridge liked to be in control and wield her authority to ensure this - a battle where she’d be unable to control anything going on and where the combatants wouldn’t care about her Ministerial credentials would be the exact opposite of a situation she’d want.
“Dolores has what amounts to a phobia of beings that are not quite, or wholly, human. Her distaste for the half-giant Hagrid, and her terror of centaurs, reveal a terror of the unknown and the wild. She is an immensely controlling person, and all who challenge her authority and world-view must, in her opinion, be punished.”
- Dolores Umbridge (Pottermore)
She’d be entirely out of her element in a battle. Her position would mean nothing, and no one would consider her an authority. They’d fight her like they would anyone else.