I suggest 'unregistered micro-business'.

The question asks for a single word, but considers the two-word examples informal business and sole proprietorship, so in the lack of finding a single word I submit this.

This type of informal and itinerant business is called street trading. Lexico has

street trader
_NOUN

A person who sells something in the street, either from a stall or van or with their goods laid out on the pavement.

---

The term is used in Africa. Quoting from JSTOR (part of ITHAKA)

Street Traders and the Emerging Spaces for Urban Voice and Citizenship in African Cities

As informal commerce has grown to become the lifeblood of African cities, street trade – among the largest sub-groups in the informal economy – has become a visible but contested domain. Yet the increase in street traders has not been accompanied by a corresponding improvement in their status as citizens or in their political influence.

And quoting from a Working Paper PDF, it is associated with informal jobs and the unregulated economy.

Legal Pluralism and Development: Street-trade and regulatory reform in Tanzania

The rapid growth of street-trade in sub-Saharan Africa (SSA) is deeply rooted in wider economic and policy trends. Characteristic of African cities for many years (Potts 2007), the numbers of informal workers and the economic and social significance of the informal economy have grown rapidly since the early 1990s (Skinner 2008a) - associated with stagnant agriculture (Bryceson 2004), the decline in formal jobs which accompanied structural adjustment, rapidly falling foreign investment levels, stagnant domestic investment and declining manufacturing (Jenkins et al 2007).

So it can be seen that street trading is associated with the poorer parts of society.

How about peddler

one who offers merchandise (such as fresh produce) for sale along the street or from door to door (Websters)

A peddler is someone who goes from place to place in order to sell something. (Collins)

If that's too formal for you, then perhaps huckster

1. a retailer of small articles, esp. a peddler of fruits and vegetables; hawker
2. a person who employs showy methods to effect a sale, win votes, etc the crass methods of political hucksters
3. a cheaply mercenary person (Collins)

I am looking for a word that describes individual economic activity in third world countries that supports a person and his family's livelihood.

This is usually described as "subsistence-level employment"

MW

subsistence level (noun)

: a level of income that provides only enough money for basic needs

living below (the) subsistence level

See also,

Cambridge dictionary:

subsistence noun [ U ] formal

the state of having what you need in order to stay alive, but no more:

The money is intended to provide a basic subsistence and should not be paid to someone who receives other income.

The family was living at subsistence level.

Hawker

A hawker is someone that does not have a permanent shop and carries their merchandise to sell on the streets. According to Dictionary.com (in the second sense):

a person who offers goods for sale by shouting his or her wares in the street or going from door to door; peddler.

A cottage industry is a special kind of business wherein people work out of their homes for a lack of better real estate. According to Investopedia,

A cottage industry is a small-scale, decentralized manufacturing business often operated out of a home rather than a purpose built facility.

A day labourer does not work for themselves but is a poor, contractual hire receiving not more than a day's work at a time. According to Cambridge,

a person who is employed for one day at a time

A word used to describe people working like this is "hand-to-mouth", describing one that makes barely enough to survive the day.