The 5e rules for multiclassing (PHB p. 164) state that your spells known and prepared are based on your class level for each class, but your spell slots are based on your combined levels, which may result in you having spell slots of higher level than any of the spells you know. They provide an example for Ranger 4 / Wizard 3. With respect to Wizard spells known it says:
As a 3rd level wizard, you know three wizard cantrips, and your spellbook contains ten wizard spells, two of which (the two you gained when you reached 3rd level as a wizard) can be 2nd level spells. If your Intelligence is 16, you can prepare six wizard spells from your spellbook.
This combination counts as a 5th level caster for determining spell slots:
you have four 1st level slots, three 2nd level slots, and two 3rd level slots. However, you don’t know any 3rd level spells….You can use the spell slots of those (higher levels than you have spells) slots to cast the spells you do know — and potentially enhance their effects.
The rules for copying a spell into your spellbook (PHB P. 114 sidebar) require you to have spell slots of that level.
When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a level for which you have spell slots and if you can spare the time to decipher and copy it.
RAW then, it appears that the character, who does have spell slots for every level, would be permitted to copy any wizard spell she finds into her spellbook. This comes down to whether “for which you have spell slots” above must be interpreted as “for which you would have spell slots if you were a single-class wizard of your wizard level” There is evidence (see below) that it must be interpreted as such for learning and preparing spells, but the multiclass rules are silent on copying spells, so that leaves the scribing answer as a “maybe”.
The spell preparation rules in the Wizard section (PHB p. 114) state that you:
choose a number of spells from your spellbook equal to your Intelligence modifier + your wizard level (minimum of one spell). The spells must be of a level for which you have spell slots.
The learning spells section states:
Each time you gain a wizard level, you can add two wizard spells of your choice to your spellbook. Each of these spells must be of a level for which you have spell slots.
Given the example in the first quote of which spells the multiclassed wizard will have learned at wizard level 3, it is clear that “for which you have spell slots” in the final quote above must be read as “for which you would have spell slots if you were a single-class wizard of your wizard level”.
Thus we must conclude that multiclass rule statement that “You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class,” means that we must re-interpret the phrase “for which you have spell slots” in (at least) these two contexts. The multiclass rules remain silent as to whether we must also reinterpret that phrase in other contexts, such as the rule for copying spells.