A single verb that expresses the idea of "making do" is accontentarsi (to be content with something/to make oneself be content). The adjective it stems from is contento (happy, content). The non-reflexive verb accontentare can be translated as "to satisfy."

Me lo avete chiesto voi, eh, quindi io vi accontento.

You asked me for it, huh, so I will satisfy you.

Caption 6, Marika spiega - I verbi cavare e togliere

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You are giving someone what they want. You are making them happy.

Making do

The reflexive verb accontentarsi implies that something may not be up to our highest expectations or greatest desires, but it will do, because we decide to accept it. We settle for it. 

Quando ho molto tempo, preferisco mangiare frutta, latte e cereali;

When I have lots of time, I prefer to eat fruit, milk and cereal;

quando ho poco tempo,

when I have little time,

mi accontento del classico caffè e del cornetto o brioche.

I make do with a classic espresso and croissant or brioche.

Captions 20-23, Adriano - Giornata

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Adding some color

The verb accontentarsi has a lot of information in it, but Italians have an expression that enhances it even further. Italy, being a Roman Catholic country historically, is not lacking in monasteries and convents. While in English, "convent" tends to be understood as a convent of nuns, in Italian, un convento implies a religious community and may be either di suore (of nuns  = convent) or di frati (of monks = monastery). Many conventi around Italy offer hospitality to travelers, but the food that is served is the humble and simple fare the monks or nuns are served. And of course, they don't complain about it. 

So let's say someone asks you to stay for dinner on the spur of the moment and doesn't have anything special to offer. 

Se ti accontenti di quel che passa il convento, sei il benvenuto (if you make do with what the convent is serving [what we have on hand], you are welcome to stay for dinner).

But the expression is used outside of the realm of food, too. In this clip, we're talking about what kind of work one can get.

Guardi che Gigi c'ha pure due lauree.

Look, Gigi even has two degrees.

-E fa il deejay?

-And he is deejaying?

-E questo passa il convento.

-Well, that's what the convent offers [beggars can't be choosers].

Captions 13-15, La Ladra EP. 8 - Il momento giusto

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In an episode of Volare, the expression is used rather vulgarly, referring to a woman. But now, when you watch the video, you'll understand what's behind this expression.

Me so' [romanesco: mi sono] accontentato di quel che passava il convento.

I made do with what the convent was serving.

Caption 40, Volare - La grande storia di Domenico Modugno - Ep. 1

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A couple of additional examples:

-I'm talking to my husband about lunch:

Vuoi anche un secondo o ti accontenti di un piatto di pasta e un'insalata? (do you want a second course or are you happy with pasta and salad)?

-My boss asks me:

Mi puoi fare una bozza per domani (can you give me a rough draft by tomorrow)?

I answer:

Non so se ce la faccio, ma farò del mio meglio per accontentarti (I don't know if I'll be able to, but I'll do my best to satisfy you).