The Đại Nam Imperial Army is divided into (I'll do my best to try and transliterate it into English)

Standing Army

Infantry:

Organization:

Regiment: 2,500 men (5 Vanguards)

Vanguard: 500 men (10 Decarion) (Has 2 cannons and 200 muskets each)

Centurion: 50 men

Pentarion: 5 men

No mention of numbers aside from that there's 40,000 in the Capital and that there are 120,000 total at any time (which might include sailors in the navy). This makes sense as 40k Number of sailors other branches=Around 120,000

Elephant Troops

Wikipedia says that the Elephant Troops has 1 Field Regiments with 5 Vanguards in the Capital with multiple independent Vanguards in other provinces.

The War Elephants count for the era in game according to the Emperor's reign Vietnamese wikipedia page:

Capital has 150, Tonkin has 110, Gia Dinh has 70, Quảng Nam has 35, Bình Định has 30, Nghệ An has 21, Quảng Bình, Quảng Ngãi, Thanh Hóa each has 15; Quảng Trị, Phú Yên, Bình Thuận and Ninh Bình each has 7.

My estimate would be around ~6000 people in the Elephant Troops corp.

Cavalry

No mentionings on numbers or organization, only that it is more of a guard force and mainly for ceremonial purposes.

Estimation of numbers: ~2500 at best

Artillery

Same organization as the Infantry, but each Vanguard has 10 cannons instead of 2.

Estimated Numbers: Rather specialized so I'd say ~7500 or 3 Regiments

Marines

Wikipedia mentioned 15 Vanguards in 3 Regiments so that'd be 7,500 marines.

Navy

One Naval Division was reported to have 9 European vessels armed with 60 guns and 5 vessels with 50 guns, 40 gunboats with 16 guns, 100 junks, 119 galleys and 365 smaller boats.

Vietnamese Wikipedia mentioned "a few" European style ships and 200 gunboats with 16-22 guns and 600 junks galleys or other small boats (not counting transports) which would be close to the report above, for a naval strength of:

9x 60-guns (European style) Ships

5x 50-guns (European style) Ships

40-50x 16-guns (Eastern-style) Ships (with a few European sloops, probably less than 2 digits)

100 junks, 119 galleys, and 365 smaller boats. (Junks and Galleys are small, fast and nimble carrying a single cannon)

Based off of Tay Son sailor numbers, the number of sailors in the navy would be around 60,000.

Levy Soldiers

They are farmers for 2/3 of the year and soldiers for 1/3 of the year. They're provincial garrisons and number at 150,000 in the early 1800s in total and "60,000 in Tonkin" in 1880.

Note: An army of 120,000 standing soldiers (admittedly only 40,000 are actual infantry) might seem enormous but considering this Battle which occurred 40 years prior where the Vietnamese had 100,000 soldiers against the Qing's 40,000-220,000 soldiers. Asia has always had massive armies.