Invoicing between partners is in general not allowed within Horizon Europe. According to the Annotated Model Grant Agreement of Horizon Europe, purchases or invoicing between beneficiaries is not considered eligible as there is the risk of double funding or charging commercial profit margins. This rule is a principle but there might be exceptions, such as when the partner is the sole supplier of a specific good or service. However this has to be duly demonstrated.
Note that invoicing between partners in Horizon Europe is not allowed in the framework of a project, meaning that a partner/beneficiary cannot claim an invoice issued by another partner. But a private commercial relationship between two partners/beneficiaries outside of the grant activity is obviously allowed and invoicing allowed.
Other article of interest: Is subcontracting between partners allowed in Horizon Europe?
General > Article 6.2.C Purchase costs
Specific cases (purchases):
Purchases between beneficiaries — Are in principle not accepted. If a beneficiary needs supplies from another beneficiary, it is the latter beneficiary that should charge them to the action as cost. (Otherwise there is the risk that the grant is used to charge commercial profit margins.) Purchases between beneficiaries will only be accepted in exceptional and properly justified cases (e.g. beneficiary A is the usual supplier of beneficiary B for a generic consumable that beneficiary B needs for the action).