Distribution missions, like all missions require a minimum level of standing with the faction, corporation or agent. You can always request a level 1 mission but need +1, +3 and +5 respectively for level 2, 3 and 4 missions.
If you decline 2 missions in a 4 hour period, you will take a standing hit. Best to work in a mission hub with several agents in close proximity so you have options if you are offered a mission you don’t want to accept.
Social skills: Social and Connections will improve standing allowing you to run higher level missions; Negotiation and Distribution Connections will improve the rewards.
A T1 exploration frigate is a good choice for level 1 missions - you could end up hauling, mining or fighting and these ships can do it all.
For higher level distribution missions use one of the fast haulers - Nereus, Wreathe, etc… They can easily be fit to accommodate standard cargo and the large cargo you can get with storyline missions can be split and moved in pieces. A Blockade Runner also works well if you have the skills to fly it.
These missions are a good way to grind standing but, probably not the best choice for getting rich. Pirate faction LP tends to be quite a bit more valuable than LP from the Empire factions so consider running L4 distribution missions for SOE out of Gicodel or Thukker Mix out of Totkubad or Oremmulf.