Any summons or familiars need to be approved by the DM before you introduce them to the table, so that the token image you want to use can be imported and preparation can be made for their abilities and features. There will be limits on which creatures are suitable for particular campaign, or limits placed due to the game type (such as one-shots) to keep turns quick and games fun.
Familiars 
Check a specific form for a familiar with your DM. In certain game types (mostly weekly tables, though NOT Escape or Dark Order) homebrew familiar summoning scrolls can be purchased from the Wandering Merchant.
Mounts
Beasts can be purchased and equipped as mounts with a maximum Challenge Rating based on your player level divided by 2 (up to a cap of 8).
Standard cost for beast mounts is (1d100 + 100) multiplied by the CR value. Barding (mount armor) can be purchased at 4 times the price of normal character armor.
Summons
In general you can’t summon a creature that has Legendary Actions or Legendary Resistances (even if you don’t intend to use them) unless your DM has specifically approved it. Other limitations on the possible forms may apply – for example only certain books may be allowed for Escape games, so please check any creatures with your DM.
Unless specifically named in a spell or feature (like Homunculus servant, Find Familiar) creatures listed on DDB with a Challenge Rating of “zero” cannot be summoned or used – DDB doesn’t assign CR to creatures that only exist inside specific campaign or item scenarios, so they’re not meant to be accessible to players outside of those contexts. Examples of invalid summons are the Avatar of Death from the Deck of Many Things.
Wildshape, Shapechange & Morph
Similar to summoning, you can’t change into a creature that has Legendary Actions or Legendary Resistances (even if you don’t intend to use them). Other limitations on the possible forms may apply (only certain books are allowed for Escape games for example), check a specific creature with your DM.
Items & Attunement
If it makes reasonable sense for a mount or familiar to be able to use an item (intelligence level, right form to wield or carry) then they can equip items from your inventory that you give them. If the item requires attunement (and the DM says your creature is intelligent enough to do so, probably INT 6 at least) then they can spend 1 hour to attune as normal – but this will use one of your character’s attunement slots, they don’t have any slots of their own.
Token Limits
To prevent tables getting bogged down with dozens of tokens each taking their own turns, the general rule is that:
- Total CR of any additional tokens on the board per player (familiars, summons etc) must be less than or equal to your character level
- Maximum count of all tokens on the table per player is 6 to keep things moving